Are there any two numbers such that multiplying them together is the same as putting their digits next to...











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I have two natural numbers, A and B, such that A * B = AB.



Do any such numbers exist? For example, if 20 and 18 were such numbers then 20 * 18 = 2018.



From trying out a lot of different combinations, it seems as though putting the digits of the numbers together always overestimates, but I have not been able to prove this yet.



So, I have 3 questions:




  1. Does putting the digits next to each other always overestimate? (If so, please prove this.)

  2. If it does overestimate, is there any formula for computing by how much it will overestimate in terms of the original inputs A and B? (A proof that there's no such formula would be wonderful as well.)

  3. Are there any bases (not just base 10) for which there are such numbers? (Negative bases, maybe?)










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  • 5




    If $B$ has $b$ digits then $B < 10^b$ but $AB > 10^b A$. This argument applies in any (positive) base.
    – Qiaochu Yuan
    Dec 8 at 0:43








  • 1




    Yes. I admit that was a little unclear given that you're using it to mean concatenation.
    – Qiaochu Yuan
    Dec 8 at 0:48






  • 3




    @TobyMak that's still overestimating, just like 20 * 18 < 2018
    – Pro Q
    Dec 8 at 0:48






  • 4




    Just a side note: as multiplication of numbers is commutative, those we would need A=B, because otherwise the concatenations A.B and B.A would yield different results whereas AB and BA are equal.
    – Philipp Imhof
    Dec 8 at 18:31






  • 3




    @PhilippImhof Not necessarily, since you're not asking that the same holds for the reversed order. For instance change a bit the quest and suppose to look for $A, B$ such that $2Acdot B=A*B$, where * is the concatenation and $cdot$ the multiplication. Then your reasoning should apply again, but $A=3, B=6$ is a solution.
    – Del
    Dec 9 at 11:16















up vote
30
down vote

favorite
5












I have two natural numbers, A and B, such that A * B = AB.



Do any such numbers exist? For example, if 20 and 18 were such numbers then 20 * 18 = 2018.



From trying out a lot of different combinations, it seems as though putting the digits of the numbers together always overestimates, but I have not been able to prove this yet.



So, I have 3 questions:




  1. Does putting the digits next to each other always overestimate? (If so, please prove this.)

  2. If it does overestimate, is there any formula for computing by how much it will overestimate in terms of the original inputs A and B? (A proof that there's no such formula would be wonderful as well.)

  3. Are there any bases (not just base 10) for which there are such numbers? (Negative bases, maybe?)










share|cite|improve this question


















  • 5




    If $B$ has $b$ digits then $B < 10^b$ but $AB > 10^b A$. This argument applies in any (positive) base.
    – Qiaochu Yuan
    Dec 8 at 0:43








  • 1




    Yes. I admit that was a little unclear given that you're using it to mean concatenation.
    – Qiaochu Yuan
    Dec 8 at 0:48






  • 3




    @TobyMak that's still overestimating, just like 20 * 18 < 2018
    – Pro Q
    Dec 8 at 0:48






  • 4




    Just a side note: as multiplication of numbers is commutative, those we would need A=B, because otherwise the concatenations A.B and B.A would yield different results whereas AB and BA are equal.
    – Philipp Imhof
    Dec 8 at 18:31






  • 3




    @PhilippImhof Not necessarily, since you're not asking that the same holds for the reversed order. For instance change a bit the quest and suppose to look for $A, B$ such that $2Acdot B=A*B$, where * is the concatenation and $cdot$ the multiplication. Then your reasoning should apply again, but $A=3, B=6$ is a solution.
    – Del
    Dec 9 at 11:16













up vote
30
down vote

favorite
5









up vote
30
down vote

favorite
5






5





I have two natural numbers, A and B, such that A * B = AB.



Do any such numbers exist? For example, if 20 and 18 were such numbers then 20 * 18 = 2018.



From trying out a lot of different combinations, it seems as though putting the digits of the numbers together always overestimates, but I have not been able to prove this yet.



So, I have 3 questions:




  1. Does putting the digits next to each other always overestimate? (If so, please prove this.)

  2. If it does overestimate, is there any formula for computing by how much it will overestimate in terms of the original inputs A and B? (A proof that there's no such formula would be wonderful as well.)

  3. Are there any bases (not just base 10) for which there are such numbers? (Negative bases, maybe?)










share|cite|improve this question













I have two natural numbers, A and B, such that A * B = AB.



Do any such numbers exist? For example, if 20 and 18 were such numbers then 20 * 18 = 2018.



From trying out a lot of different combinations, it seems as though putting the digits of the numbers together always overestimates, but I have not been able to prove this yet.



So, I have 3 questions:




  1. Does putting the digits next to each other always overestimate? (If so, please prove this.)

  2. If it does overestimate, is there any formula for computing by how much it will overestimate in terms of the original inputs A and B? (A proof that there's no such formula would be wonderful as well.)

  3. Are there any bases (not just base 10) for which there are such numbers? (Negative bases, maybe?)







natural-numbers






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asked Dec 8 at 0:30









Pro Q

325313




325313








  • 5




    If $B$ has $b$ digits then $B < 10^b$ but $AB > 10^b A$. This argument applies in any (positive) base.
    – Qiaochu Yuan
    Dec 8 at 0:43








  • 1




    Yes. I admit that was a little unclear given that you're using it to mean concatenation.
    – Qiaochu Yuan
    Dec 8 at 0:48






  • 3




    @TobyMak that's still overestimating, just like 20 * 18 < 2018
    – Pro Q
    Dec 8 at 0:48






  • 4




    Just a side note: as multiplication of numbers is commutative, those we would need A=B, because otherwise the concatenations A.B and B.A would yield different results whereas AB and BA are equal.
    – Philipp Imhof
    Dec 8 at 18:31






  • 3




    @PhilippImhof Not necessarily, since you're not asking that the same holds for the reversed order. For instance change a bit the quest and suppose to look for $A, B$ such that $2Acdot B=A*B$, where * is the concatenation and $cdot$ the multiplication. Then your reasoning should apply again, but $A=3, B=6$ is a solution.
    – Del
    Dec 9 at 11:16














  • 5




    If $B$ has $b$ digits then $B < 10^b$ but $AB > 10^b A$. This argument applies in any (positive) base.
    – Qiaochu Yuan
    Dec 8 at 0:43








  • 1




    Yes. I admit that was a little unclear given that you're using it to mean concatenation.
    – Qiaochu Yuan
    Dec 8 at 0:48






  • 3




    @TobyMak that's still overestimating, just like 20 * 18 < 2018
    – Pro Q
    Dec 8 at 0:48






  • 4




    Just a side note: as multiplication of numbers is commutative, those we would need A=B, because otherwise the concatenations A.B and B.A would yield different results whereas AB and BA are equal.
    – Philipp Imhof
    Dec 8 at 18:31






  • 3




    @PhilippImhof Not necessarily, since you're not asking that the same holds for the reversed order. For instance change a bit the quest and suppose to look for $A, B$ such that $2Acdot B=A*B$, where * is the concatenation and $cdot$ the multiplication. Then your reasoning should apply again, but $A=3, B=6$ is a solution.
    – Del
    Dec 9 at 11:16








5




5




If $B$ has $b$ digits then $B < 10^b$ but $AB > 10^b A$. This argument applies in any (positive) base.
– Qiaochu Yuan
Dec 8 at 0:43






If $B$ has $b$ digits then $B < 10^b$ but $AB > 10^b A$. This argument applies in any (positive) base.
– Qiaochu Yuan
Dec 8 at 0:43






1




1




Yes. I admit that was a little unclear given that you're using it to mean concatenation.
– Qiaochu Yuan
Dec 8 at 0:48




Yes. I admit that was a little unclear given that you're using it to mean concatenation.
– Qiaochu Yuan
Dec 8 at 0:48




3




3




@TobyMak that's still overestimating, just like 20 * 18 < 2018
– Pro Q
Dec 8 at 0:48




@TobyMak that's still overestimating, just like 20 * 18 < 2018
– Pro Q
Dec 8 at 0:48




4




4




Just a side note: as multiplication of numbers is commutative, those we would need A=B, because otherwise the concatenations A.B and B.A would yield different results whereas AB and BA are equal.
– Philipp Imhof
Dec 8 at 18:31




Just a side note: as multiplication of numbers is commutative, those we would need A=B, because otherwise the concatenations A.B and B.A would yield different results whereas AB and BA are equal.
– Philipp Imhof
Dec 8 at 18:31




3




3




@PhilippImhof Not necessarily, since you're not asking that the same holds for the reversed order. For instance change a bit the quest and suppose to look for $A, B$ such that $2Acdot B=A*B$, where * is the concatenation and $cdot$ the multiplication. Then your reasoning should apply again, but $A=3, B=6$ is a solution.
– Del
Dec 9 at 11:16




@PhilippImhof Not necessarily, since you're not asking that the same holds for the reversed order. For instance change a bit the quest and suppose to look for $A, B$ such that $2Acdot B=A*B$, where * is the concatenation and $cdot$ the multiplication. Then your reasoning should apply again, but $A=3, B=6$ is a solution.
– Del
Dec 9 at 11:16










7 Answers
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up vote
8
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accepted










If $B$ has $n$ digits then $10^{n-1} le B <10^n$ and we want $AB = 10^nA + B$ or



But $B<10^n$ so $AB < 10^nA le 10^nA + B$



So 1) Yes over compensation always



2) by $10^nA + B - AB = 10^{[log_{10}(B)]+1}A + B - AB$



3) The same argument applies to any base $> 1$






share|cite|improve this answer























  • By AB for part (2), do you mean A*B?
    – Pro Q
    Dec 8 at 7:55










  • Also, I believe (1) should include the case where A = B = 0
    – Pro Q
    Dec 8 at 8:02






  • 1




    I don't think we should include $A=B=0$ because $00$ isn't a valid expression. I took it as a given that $A ge 1$. But if you wish to include that as a case you may. And yes $AB$ is the product of $A$ and $B$.
    – fleablood
    Dec 8 at 16:59






  • 1




    It's perfectly consistant. I always used AB to mean A times B and i never used it to mean anything else. I never used any notation for concatenation.
    – fleablood
    Dec 9 at 5:01






  • 1




    Of course. The left hand side is $AB$ is multiplying the two digits. And $10^nA + B$ the right hand side is concatinating.
    – fleablood
    Dec 10 at 2:22


















up vote
41
down vote














I have two natural numbers, $A$ and $B$, such that $A times B = AB$.



Do any such numbers exist? For example, if $20$ and $18$ were such numbers then $20 times 18 = 2018$.




Lets put aside the trivial answer $A=0$ and $B=0$ and consider both $A, B>0$.



You want numbers such that $Atimes B = Atimes10^k + B$ where $k$ is the number of digits of $B$, that is with $10^{k-1}leq B < 10^k$. So you need $B=10^k+dfrac{B}{A}$ with $B<10^k$. From which results $B>10^k$ and $B<10^k$.



So if there's no mistake there, the answer is no.






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  • 6




    This is really nice.
    – Randall
    Dec 8 at 1:34


















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27
down vote













There is the pathological example $A=B=0$.



For the rest:




Let $B$ have $m$ digits. We have $AB= A*10^m+B$ We want $AB=A* B$,



We have $$A*10^m+B-A*B = A*(10^m-B)+B>B,$$
because $10^m>B$ as $B$ has $m$ digits. So you always overestimate by at least $B$.



From this its also clear, that the result is independent of the chosen
base.








Let me generealise a bit:

If we allow $A$ to be negative, we need to change the condition to $AB=A*10^m-B$.

This leads to
$$A=frac B {10^m-B}, $$
but then the right hand side is positive and again we get a contradiction.



So the last possibility is $B<0$. But then we first need to define $AB$.

A natural way to do this would be $A(-B)$.

Then we have for $A>0$: $AB=A*10^m-B$ which implies
$$A=frac B {10^m-B}. $$
This time, there is no contradiction because of sign issues.

However, now $-B>0$, hence $|10^m-B|>|B|$, so the right hand side is no integer.



The analogous argument gives also a contradiction for $A,B>0$.



So even in the more generalised setting, the answer is no.






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    up vote
    4
    down vote













    I will use $*$ for your concatenation operation, and $cdot$ for true multiplication.



    1) If you allow for leading zeroes to be ingored, then $0*0=00=0$. Notice that for all pairs of non-zero natural numbers, $acdot bleq acdot10^{lceillog_{10}(b)rceil}$ but that $a*b>acdot10^{lceillog_{10}(b)rceil}$. If $a=0$ and $bneq 0$ and we ignore leading zeroes, then we are still an overestimation, and if $aneq0$ and $b=0$ then we are still an overstimation.



    2) Based on my crude estimates above, yes, there is a way of putting bounds on the size of the overestimation, but I don't know if you can do much better honestly.



    3) Also based on my crude estimations, if you replace the logarithms with different bases I'm fairly sure that this shows no base greater than 1 works. Base 1 itself actually has both types of behaviour; $1*11=111>11$, but $111*111=111111<111111111$. Additionally, you have an example of equality in $11*11=1111$. There is of course the issue that $0$ is a strange object to try and work with in base 1, so let's just ignore that for now...



    I can't muster the strength of will to try and prove anything for negative bases. I suspect that negative bases less than $-1$ will fail, but it is easy to see that in base $-1$ there are trivial representations that will also give you equality; $11*11=1111$ where everything in sight is $0$ in base 10.






    share|cite|improve this answer



















    • 2




      Those are not base-1 numbers, as in any base b, the allowed digits are from 0 to b-1
      – Ben Voigt
      Dec 9 at 3:45






    • 3




      @BenVoigt: In unary numeral system, you can use any arbitrary symbol for tallying. If you insist, you can use $0$ as repeated symbol. So zero would be represented with an empty string (not very convenient...), one with $0$, two with $00$ and four with $0000$.
      – Eric Duminil
      Dec 9 at 16:13








    • 1




      @EricDuminil: That's a fine system and "unary" is a good name for it, but not "base-1", since it has no relationship to positional number systems. It does not qualify as an answer to OP's final question "Are there any bases for which there are such numbers?"
      – Ben Voigt
      Dec 10 at 0:31




















    up vote
    1
    down vote













    So it was impossible with natural numbers (see other answers)
    But if you willing to bend the rules a bit, by making that when decimal numbers are involved $A.a times B.b = AB.ab$



    then you can find



    $$x.99999999999... times 9.9999999999... = x9.99999999999...$$



    or more compactly



    $$x.(9) times 9.(9) = x9.(9)$$



    this is possible because $x.(9) = x + 1$



    and for an example if $x=4$



    $5 times 10 = 50 Leftrightarrow 4.(9) times 9.(9) = 49.(9)$






    share|cite|improve this answer




























      up vote
      0
      down vote













      (1) A slightly different method:



      Let $k>1$ be an an arbitrary base. We know that $lceillog_kBrceil$ is the number of digits in $B$ in base $k$.



      We want to prove if there exists a solution to $A*B=A*k^{lceil log_{k}Brceil}$+B.



      Isolate $A$ and $B$, $::1-frac{1}{A}=frac{k^{lceil log_{k}Brceil}}{B}$.
      For positive $A$, we have that $1-frac{1}{A} < 1$.



      Recall that by the definition of logarithm, $k^{log_kB}=B$. $:$ Since$lceil xrceilgeq x$, we know $frac{k^{lceil log_{k}Brceil}}{B}geq1$. Thus, along with the other answers, we come to a contradiction. One side of the equation is less than 1, the other at least 1.





      (3) Now what if $k<0$?



      I will first point out that there do exist integers $A$ and $B$ for which $A * B=AB$.



      We will use the following definition of negative base 10:




      A number of the form $ldots d_2d_1d_0$ with numerical value $ldots+ d_2(-10)^2+d_1(-10)^1+d_0(-10)^0$ where every $0leq d_ileq9.$




      Now consider $A=-4_{10}=-4_{-10}$ and $B=8_{10}=8_{-10}$. Discarding the negative sign, $AB=48$.



      $A*B=-32_{10}=4(-10)^1+8(-10)^0=48_{-10}$.



      Another example:



      $A=-9_{10}=-9_{-10}, B=90_{10}=90_{-10}$. $AB =990$.



      $A*B=-810_{10}=9(-10)^1+9(-10)^1+0(-10)^0=-990_{-10}$. This time we must include the negative sign, but regardless the digits are the same.



      So for your third question, provided that we let the negative sign in the concatenation be optional, we can find examples where multiplying two numbers will yield the concatenation of their digits.






      share|cite|improve this answer






























        up vote
        0
        down vote













        Here's my approach: take two natural numbers $n,m$ with $x,y$ number of digits respectively. Then in particular we can bound $$n cdot m leq (9 cdots text{($x$ times)} cdots 9) cdot (9 cdots text{($y$ times)} cdots 9) = (10^{x} - 1)(10^y - 1),$$
        so $n cdot m leq 10^{x+y} - 10^x - 10^y + 1$. Now, we can also bound
        $$nm geq (10 cdots text{($x-1$ zeros)} cdots 0)(10 cdots text{($y-1$ zeros)} cdots 0) geq 10 cdots text{($x+y-1$ zeros)} cdots 0,$$ so that $nm geq 10^{x+y-1}$.



        This is as far as I've got, there are already better answers around.






        share|cite|improve this answer























        • Absolutely right - fixed it now. Probably would've been more standard to swap the roles of $x,y$ and $n,m$ but meh shrugs
          – Stuartg98
          Dec 9 at 21:08











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        7 Answers
        7






        active

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        7 Answers
        7






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        active

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        up vote
        8
        down vote



        accepted










        If $B$ has $n$ digits then $10^{n-1} le B <10^n$ and we want $AB = 10^nA + B$ or



        But $B<10^n$ so $AB < 10^nA le 10^nA + B$



        So 1) Yes over compensation always



        2) by $10^nA + B - AB = 10^{[log_{10}(B)]+1}A + B - AB$



        3) The same argument applies to any base $> 1$






        share|cite|improve this answer























        • By AB for part (2), do you mean A*B?
          – Pro Q
          Dec 8 at 7:55










        • Also, I believe (1) should include the case where A = B = 0
          – Pro Q
          Dec 8 at 8:02






        • 1




          I don't think we should include $A=B=0$ because $00$ isn't a valid expression. I took it as a given that $A ge 1$. But if you wish to include that as a case you may. And yes $AB$ is the product of $A$ and $B$.
          – fleablood
          Dec 8 at 16:59






        • 1




          It's perfectly consistant. I always used AB to mean A times B and i never used it to mean anything else. I never used any notation for concatenation.
          – fleablood
          Dec 9 at 5:01






        • 1




          Of course. The left hand side is $AB$ is multiplying the two digits. And $10^nA + B$ the right hand side is concatinating.
          – fleablood
          Dec 10 at 2:22















        up vote
        8
        down vote



        accepted










        If $B$ has $n$ digits then $10^{n-1} le B <10^n$ and we want $AB = 10^nA + B$ or



        But $B<10^n$ so $AB < 10^nA le 10^nA + B$



        So 1) Yes over compensation always



        2) by $10^nA + B - AB = 10^{[log_{10}(B)]+1}A + B - AB$



        3) The same argument applies to any base $> 1$






        share|cite|improve this answer























        • By AB for part (2), do you mean A*B?
          – Pro Q
          Dec 8 at 7:55










        • Also, I believe (1) should include the case where A = B = 0
          – Pro Q
          Dec 8 at 8:02






        • 1




          I don't think we should include $A=B=0$ because $00$ isn't a valid expression. I took it as a given that $A ge 1$. But if you wish to include that as a case you may. And yes $AB$ is the product of $A$ and $B$.
          – fleablood
          Dec 8 at 16:59






        • 1




          It's perfectly consistant. I always used AB to mean A times B and i never used it to mean anything else. I never used any notation for concatenation.
          – fleablood
          Dec 9 at 5:01






        • 1




          Of course. The left hand side is $AB$ is multiplying the two digits. And $10^nA + B$ the right hand side is concatinating.
          – fleablood
          Dec 10 at 2:22













        up vote
        8
        down vote



        accepted







        up vote
        8
        down vote



        accepted






        If $B$ has $n$ digits then $10^{n-1} le B <10^n$ and we want $AB = 10^nA + B$ or



        But $B<10^n$ so $AB < 10^nA le 10^nA + B$



        So 1) Yes over compensation always



        2) by $10^nA + B - AB = 10^{[log_{10}(B)]+1}A + B - AB$



        3) The same argument applies to any base $> 1$






        share|cite|improve this answer














        If $B$ has $n$ digits then $10^{n-1} le B <10^n$ and we want $AB = 10^nA + B$ or



        But $B<10^n$ so $AB < 10^nA le 10^nA + B$



        So 1) Yes over compensation always



        2) by $10^nA + B - AB = 10^{[log_{10}(B)]+1}A + B - AB$



        3) The same argument applies to any base $> 1$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 8 at 5:08

























        answered Dec 8 at 1:25









        fleablood

        67.9k22684




        67.9k22684












        • By AB for part (2), do you mean A*B?
          – Pro Q
          Dec 8 at 7:55










        • Also, I believe (1) should include the case where A = B = 0
          – Pro Q
          Dec 8 at 8:02






        • 1




          I don't think we should include $A=B=0$ because $00$ isn't a valid expression. I took it as a given that $A ge 1$. But if you wish to include that as a case you may. And yes $AB$ is the product of $A$ and $B$.
          – fleablood
          Dec 8 at 16:59






        • 1




          It's perfectly consistant. I always used AB to mean A times B and i never used it to mean anything else. I never used any notation for concatenation.
          – fleablood
          Dec 9 at 5:01






        • 1




          Of course. The left hand side is $AB$ is multiplying the two digits. And $10^nA + B$ the right hand side is concatinating.
          – fleablood
          Dec 10 at 2:22


















        • By AB for part (2), do you mean A*B?
          – Pro Q
          Dec 8 at 7:55










        • Also, I believe (1) should include the case where A = B = 0
          – Pro Q
          Dec 8 at 8:02






        • 1




          I don't think we should include $A=B=0$ because $00$ isn't a valid expression. I took it as a given that $A ge 1$. But if you wish to include that as a case you may. And yes $AB$ is the product of $A$ and $B$.
          – fleablood
          Dec 8 at 16:59






        • 1




          It's perfectly consistant. I always used AB to mean A times B and i never used it to mean anything else. I never used any notation for concatenation.
          – fleablood
          Dec 9 at 5:01






        • 1




          Of course. The left hand side is $AB$ is multiplying the two digits. And $10^nA + B$ the right hand side is concatinating.
          – fleablood
          Dec 10 at 2:22
















        By AB for part (2), do you mean A*B?
        – Pro Q
        Dec 8 at 7:55




        By AB for part (2), do you mean A*B?
        – Pro Q
        Dec 8 at 7:55












        Also, I believe (1) should include the case where A = B = 0
        – Pro Q
        Dec 8 at 8:02




        Also, I believe (1) should include the case where A = B = 0
        – Pro Q
        Dec 8 at 8:02




        1




        1




        I don't think we should include $A=B=0$ because $00$ isn't a valid expression. I took it as a given that $A ge 1$. But if you wish to include that as a case you may. And yes $AB$ is the product of $A$ and $B$.
        – fleablood
        Dec 8 at 16:59




        I don't think we should include $A=B=0$ because $00$ isn't a valid expression. I took it as a given that $A ge 1$. But if you wish to include that as a case you may. And yes $AB$ is the product of $A$ and $B$.
        – fleablood
        Dec 8 at 16:59




        1




        1




        It's perfectly consistant. I always used AB to mean A times B and i never used it to mean anything else. I never used any notation for concatenation.
        – fleablood
        Dec 9 at 5:01




        It's perfectly consistant. I always used AB to mean A times B and i never used it to mean anything else. I never used any notation for concatenation.
        – fleablood
        Dec 9 at 5:01




        1




        1




        Of course. The left hand side is $AB$ is multiplying the two digits. And $10^nA + B$ the right hand side is concatinating.
        – fleablood
        Dec 10 at 2:22




        Of course. The left hand side is $AB$ is multiplying the two digits. And $10^nA + B$ the right hand side is concatinating.
        – fleablood
        Dec 10 at 2:22










        up vote
        41
        down vote














        I have two natural numbers, $A$ and $B$, such that $A times B = AB$.



        Do any such numbers exist? For example, if $20$ and $18$ were such numbers then $20 times 18 = 2018$.




        Lets put aside the trivial answer $A=0$ and $B=0$ and consider both $A, B>0$.



        You want numbers such that $Atimes B = Atimes10^k + B$ where $k$ is the number of digits of $B$, that is with $10^{k-1}leq B < 10^k$. So you need $B=10^k+dfrac{B}{A}$ with $B<10^k$. From which results $B>10^k$ and $B<10^k$.



        So if there's no mistake there, the answer is no.






        share|cite|improve this answer



















        • 6




          This is really nice.
          – Randall
          Dec 8 at 1:34















        up vote
        41
        down vote














        I have two natural numbers, $A$ and $B$, such that $A times B = AB$.



        Do any such numbers exist? For example, if $20$ and $18$ were such numbers then $20 times 18 = 2018$.




        Lets put aside the trivial answer $A=0$ and $B=0$ and consider both $A, B>0$.



        You want numbers such that $Atimes B = Atimes10^k + B$ where $k$ is the number of digits of $B$, that is with $10^{k-1}leq B < 10^k$. So you need $B=10^k+dfrac{B}{A}$ with $B<10^k$. From which results $B>10^k$ and $B<10^k$.



        So if there's no mistake there, the answer is no.






        share|cite|improve this answer



















        • 6




          This is really nice.
          – Randall
          Dec 8 at 1:34













        up vote
        41
        down vote










        up vote
        41
        down vote










        I have two natural numbers, $A$ and $B$, such that $A times B = AB$.



        Do any such numbers exist? For example, if $20$ and $18$ were such numbers then $20 times 18 = 2018$.




        Lets put aside the trivial answer $A=0$ and $B=0$ and consider both $A, B>0$.



        You want numbers such that $Atimes B = Atimes10^k + B$ where $k$ is the number of digits of $B$, that is with $10^{k-1}leq B < 10^k$. So you need $B=10^k+dfrac{B}{A}$ with $B<10^k$. From which results $B>10^k$ and $B<10^k$.



        So if there's no mistake there, the answer is no.






        share|cite|improve this answer















        I have two natural numbers, $A$ and $B$, such that $A times B = AB$.



        Do any such numbers exist? For example, if $20$ and $18$ were such numbers then $20 times 18 = 2018$.




        Lets put aside the trivial answer $A=0$ and $B=0$ and consider both $A, B>0$.



        You want numbers such that $Atimes B = Atimes10^k + B$ where $k$ is the number of digits of $B$, that is with $10^{k-1}leq B < 10^k$. So you need $B=10^k+dfrac{B}{A}$ with $B<10^k$. From which results $B>10^k$ and $B<10^k$.



        So if there's no mistake there, the answer is no.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 8 at 10:05









        Mutantoe

        558411




        558411










        answered Dec 8 at 1:02









        Jorge Adriano

        59146




        59146








        • 6




          This is really nice.
          – Randall
          Dec 8 at 1:34














        • 6




          This is really nice.
          – Randall
          Dec 8 at 1:34








        6




        6




        This is really nice.
        – Randall
        Dec 8 at 1:34




        This is really nice.
        – Randall
        Dec 8 at 1:34










        up vote
        27
        down vote













        There is the pathological example $A=B=0$.



        For the rest:




        Let $B$ have $m$ digits. We have $AB= A*10^m+B$ We want $AB=A* B$,



        We have $$A*10^m+B-A*B = A*(10^m-B)+B>B,$$
        because $10^m>B$ as $B$ has $m$ digits. So you always overestimate by at least $B$.



        From this its also clear, that the result is independent of the chosen
        base.








        Let me generealise a bit:

        If we allow $A$ to be negative, we need to change the condition to $AB=A*10^m-B$.

        This leads to
        $$A=frac B {10^m-B}, $$
        but then the right hand side is positive and again we get a contradiction.



        So the last possibility is $B<0$. But then we first need to define $AB$.

        A natural way to do this would be $A(-B)$.

        Then we have for $A>0$: $AB=A*10^m-B$ which implies
        $$A=frac B {10^m-B}. $$
        This time, there is no contradiction because of sign issues.

        However, now $-B>0$, hence $|10^m-B|>|B|$, so the right hand side is no integer.



        The analogous argument gives also a contradiction for $A,B>0$.



        So even in the more generalised setting, the answer is no.






        share|cite|improve this answer



























          up vote
          27
          down vote













          There is the pathological example $A=B=0$.



          For the rest:




          Let $B$ have $m$ digits. We have $AB= A*10^m+B$ We want $AB=A* B$,



          We have $$A*10^m+B-A*B = A*(10^m-B)+B>B,$$
          because $10^m>B$ as $B$ has $m$ digits. So you always overestimate by at least $B$.



          From this its also clear, that the result is independent of the chosen
          base.








          Let me generealise a bit:

          If we allow $A$ to be negative, we need to change the condition to $AB=A*10^m-B$.

          This leads to
          $$A=frac B {10^m-B}, $$
          but then the right hand side is positive and again we get a contradiction.



          So the last possibility is $B<0$. But then we first need to define $AB$.

          A natural way to do this would be $A(-B)$.

          Then we have for $A>0$: $AB=A*10^m-B$ which implies
          $$A=frac B {10^m-B}. $$
          This time, there is no contradiction because of sign issues.

          However, now $-B>0$, hence $|10^m-B|>|B|$, so the right hand side is no integer.



          The analogous argument gives also a contradiction for $A,B>0$.



          So even in the more generalised setting, the answer is no.






          share|cite|improve this answer

























            up vote
            27
            down vote










            up vote
            27
            down vote









            There is the pathological example $A=B=0$.



            For the rest:




            Let $B$ have $m$ digits. We have $AB= A*10^m+B$ We want $AB=A* B$,



            We have $$A*10^m+B-A*B = A*(10^m-B)+B>B,$$
            because $10^m>B$ as $B$ has $m$ digits. So you always overestimate by at least $B$.



            From this its also clear, that the result is independent of the chosen
            base.








            Let me generealise a bit:

            If we allow $A$ to be negative, we need to change the condition to $AB=A*10^m-B$.

            This leads to
            $$A=frac B {10^m-B}, $$
            but then the right hand side is positive and again we get a contradiction.



            So the last possibility is $B<0$. But then we first need to define $AB$.

            A natural way to do this would be $A(-B)$.

            Then we have for $A>0$: $AB=A*10^m-B$ which implies
            $$A=frac B {10^m-B}. $$
            This time, there is no contradiction because of sign issues.

            However, now $-B>0$, hence $|10^m-B|>|B|$, so the right hand side is no integer.



            The analogous argument gives also a contradiction for $A,B>0$.



            So even in the more generalised setting, the answer is no.






            share|cite|improve this answer














            There is the pathological example $A=B=0$.



            For the rest:




            Let $B$ have $m$ digits. We have $AB= A*10^m+B$ We want $AB=A* B$,



            We have $$A*10^m+B-A*B = A*(10^m-B)+B>B,$$
            because $10^m>B$ as $B$ has $m$ digits. So you always overestimate by at least $B$.



            From this its also clear, that the result is independent of the chosen
            base.








            Let me generealise a bit:

            If we allow $A$ to be negative, we need to change the condition to $AB=A*10^m-B$.

            This leads to
            $$A=frac B {10^m-B}, $$
            but then the right hand side is positive and again we get a contradiction.



            So the last possibility is $B<0$. But then we first need to define $AB$.

            A natural way to do this would be $A(-B)$.

            Then we have for $A>0$: $AB=A*10^m-B$ which implies
            $$A=frac B {10^m-B}. $$
            This time, there is no contradiction because of sign issues.

            However, now $-B>0$, hence $|10^m-B|>|B|$, so the right hand side is no integer.



            The analogous argument gives also a contradiction for $A,B>0$.



            So even in the more generalised setting, the answer is no.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Dec 8 at 1:16

























            answered Dec 8 at 1:02









            klirk

            2,604530




            2,604530






















                up vote
                4
                down vote













                I will use $*$ for your concatenation operation, and $cdot$ for true multiplication.



                1) If you allow for leading zeroes to be ingored, then $0*0=00=0$. Notice that for all pairs of non-zero natural numbers, $acdot bleq acdot10^{lceillog_{10}(b)rceil}$ but that $a*b>acdot10^{lceillog_{10}(b)rceil}$. If $a=0$ and $bneq 0$ and we ignore leading zeroes, then we are still an overestimation, and if $aneq0$ and $b=0$ then we are still an overstimation.



                2) Based on my crude estimates above, yes, there is a way of putting bounds on the size of the overestimation, but I don't know if you can do much better honestly.



                3) Also based on my crude estimations, if you replace the logarithms with different bases I'm fairly sure that this shows no base greater than 1 works. Base 1 itself actually has both types of behaviour; $1*11=111>11$, but $111*111=111111<111111111$. Additionally, you have an example of equality in $11*11=1111$. There is of course the issue that $0$ is a strange object to try and work with in base 1, so let's just ignore that for now...



                I can't muster the strength of will to try and prove anything for negative bases. I suspect that negative bases less than $-1$ will fail, but it is easy to see that in base $-1$ there are trivial representations that will also give you equality; $11*11=1111$ where everything in sight is $0$ in base 10.






                share|cite|improve this answer



















                • 2




                  Those are not base-1 numbers, as in any base b, the allowed digits are from 0 to b-1
                  – Ben Voigt
                  Dec 9 at 3:45






                • 3




                  @BenVoigt: In unary numeral system, you can use any arbitrary symbol for tallying. If you insist, you can use $0$ as repeated symbol. So zero would be represented with an empty string (not very convenient...), one with $0$, two with $00$ and four with $0000$.
                  – Eric Duminil
                  Dec 9 at 16:13








                • 1




                  @EricDuminil: That's a fine system and "unary" is a good name for it, but not "base-1", since it has no relationship to positional number systems. It does not qualify as an answer to OP's final question "Are there any bases for which there are such numbers?"
                  – Ben Voigt
                  Dec 10 at 0:31

















                up vote
                4
                down vote













                I will use $*$ for your concatenation operation, and $cdot$ for true multiplication.



                1) If you allow for leading zeroes to be ingored, then $0*0=00=0$. Notice that for all pairs of non-zero natural numbers, $acdot bleq acdot10^{lceillog_{10}(b)rceil}$ but that $a*b>acdot10^{lceillog_{10}(b)rceil}$. If $a=0$ and $bneq 0$ and we ignore leading zeroes, then we are still an overestimation, and if $aneq0$ and $b=0$ then we are still an overstimation.



                2) Based on my crude estimates above, yes, there is a way of putting bounds on the size of the overestimation, but I don't know if you can do much better honestly.



                3) Also based on my crude estimations, if you replace the logarithms with different bases I'm fairly sure that this shows no base greater than 1 works. Base 1 itself actually has both types of behaviour; $1*11=111>11$, but $111*111=111111<111111111$. Additionally, you have an example of equality in $11*11=1111$. There is of course the issue that $0$ is a strange object to try and work with in base 1, so let's just ignore that for now...



                I can't muster the strength of will to try and prove anything for negative bases. I suspect that negative bases less than $-1$ will fail, but it is easy to see that in base $-1$ there are trivial representations that will also give you equality; $11*11=1111$ where everything in sight is $0$ in base 10.






                share|cite|improve this answer



















                • 2




                  Those are not base-1 numbers, as in any base b, the allowed digits are from 0 to b-1
                  – Ben Voigt
                  Dec 9 at 3:45






                • 3




                  @BenVoigt: In unary numeral system, you can use any arbitrary symbol for tallying. If you insist, you can use $0$ as repeated symbol. So zero would be represented with an empty string (not very convenient...), one with $0$, two with $00$ and four with $0000$.
                  – Eric Duminil
                  Dec 9 at 16:13








                • 1




                  @EricDuminil: That's a fine system and "unary" is a good name for it, but not "base-1", since it has no relationship to positional number systems. It does not qualify as an answer to OP's final question "Are there any bases for which there are such numbers?"
                  – Ben Voigt
                  Dec 10 at 0:31















                up vote
                4
                down vote










                up vote
                4
                down vote









                I will use $*$ for your concatenation operation, and $cdot$ for true multiplication.



                1) If you allow for leading zeroes to be ingored, then $0*0=00=0$. Notice that for all pairs of non-zero natural numbers, $acdot bleq acdot10^{lceillog_{10}(b)rceil}$ but that $a*b>acdot10^{lceillog_{10}(b)rceil}$. If $a=0$ and $bneq 0$ and we ignore leading zeroes, then we are still an overestimation, and if $aneq0$ and $b=0$ then we are still an overstimation.



                2) Based on my crude estimates above, yes, there is a way of putting bounds on the size of the overestimation, but I don't know if you can do much better honestly.



                3) Also based on my crude estimations, if you replace the logarithms with different bases I'm fairly sure that this shows no base greater than 1 works. Base 1 itself actually has both types of behaviour; $1*11=111>11$, but $111*111=111111<111111111$. Additionally, you have an example of equality in $11*11=1111$. There is of course the issue that $0$ is a strange object to try and work with in base 1, so let's just ignore that for now...



                I can't muster the strength of will to try and prove anything for negative bases. I suspect that negative bases less than $-1$ will fail, but it is easy to see that in base $-1$ there are trivial representations that will also give you equality; $11*11=1111$ where everything in sight is $0$ in base 10.






                share|cite|improve this answer














                I will use $*$ for your concatenation operation, and $cdot$ for true multiplication.



                1) If you allow for leading zeroes to be ingored, then $0*0=00=0$. Notice that for all pairs of non-zero natural numbers, $acdot bleq acdot10^{lceillog_{10}(b)rceil}$ but that $a*b>acdot10^{lceillog_{10}(b)rceil}$. If $a=0$ and $bneq 0$ and we ignore leading zeroes, then we are still an overestimation, and if $aneq0$ and $b=0$ then we are still an overstimation.



                2) Based on my crude estimates above, yes, there is a way of putting bounds on the size of the overestimation, but I don't know if you can do much better honestly.



                3) Also based on my crude estimations, if you replace the logarithms with different bases I'm fairly sure that this shows no base greater than 1 works. Base 1 itself actually has both types of behaviour; $1*11=111>11$, but $111*111=111111<111111111$. Additionally, you have an example of equality in $11*11=1111$. There is of course the issue that $0$ is a strange object to try and work with in base 1, so let's just ignore that for now...



                I can't muster the strength of will to try and prove anything for negative bases. I suspect that negative bases less than $-1$ will fail, but it is easy to see that in base $-1$ there are trivial representations that will also give you equality; $11*11=1111$ where everything in sight is $0$ in base 10.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Dec 8 at 0:57

























                answered Dec 8 at 0:43









                RandomMathGuy

                1872




                1872








                • 2




                  Those are not base-1 numbers, as in any base b, the allowed digits are from 0 to b-1
                  – Ben Voigt
                  Dec 9 at 3:45






                • 3




                  @BenVoigt: In unary numeral system, you can use any arbitrary symbol for tallying. If you insist, you can use $0$ as repeated symbol. So zero would be represented with an empty string (not very convenient...), one with $0$, two with $00$ and four with $0000$.
                  – Eric Duminil
                  Dec 9 at 16:13








                • 1




                  @EricDuminil: That's a fine system and "unary" is a good name for it, but not "base-1", since it has no relationship to positional number systems. It does not qualify as an answer to OP's final question "Are there any bases for which there are such numbers?"
                  – Ben Voigt
                  Dec 10 at 0:31
















                • 2




                  Those are not base-1 numbers, as in any base b, the allowed digits are from 0 to b-1
                  – Ben Voigt
                  Dec 9 at 3:45






                • 3




                  @BenVoigt: In unary numeral system, you can use any arbitrary symbol for tallying. If you insist, you can use $0$ as repeated symbol. So zero would be represented with an empty string (not very convenient...), one with $0$, two with $00$ and four with $0000$.
                  – Eric Duminil
                  Dec 9 at 16:13








                • 1




                  @EricDuminil: That's a fine system and "unary" is a good name for it, but not "base-1", since it has no relationship to positional number systems. It does not qualify as an answer to OP's final question "Are there any bases for which there are such numbers?"
                  – Ben Voigt
                  Dec 10 at 0:31










                2




                2




                Those are not base-1 numbers, as in any base b, the allowed digits are from 0 to b-1
                – Ben Voigt
                Dec 9 at 3:45




                Those are not base-1 numbers, as in any base b, the allowed digits are from 0 to b-1
                – Ben Voigt
                Dec 9 at 3:45




                3




                3




                @BenVoigt: In unary numeral system, you can use any arbitrary symbol for tallying. If you insist, you can use $0$ as repeated symbol. So zero would be represented with an empty string (not very convenient...), one with $0$, two with $00$ and four with $0000$.
                – Eric Duminil
                Dec 9 at 16:13






                @BenVoigt: In unary numeral system, you can use any arbitrary symbol for tallying. If you insist, you can use $0$ as repeated symbol. So zero would be represented with an empty string (not very convenient...), one with $0$, two with $00$ and four with $0000$.
                – Eric Duminil
                Dec 9 at 16:13






                1




                1




                @EricDuminil: That's a fine system and "unary" is a good name for it, but not "base-1", since it has no relationship to positional number systems. It does not qualify as an answer to OP's final question "Are there any bases for which there are such numbers?"
                – Ben Voigt
                Dec 10 at 0:31






                @EricDuminil: That's a fine system and "unary" is a good name for it, but not "base-1", since it has no relationship to positional number systems. It does not qualify as an answer to OP's final question "Are there any bases for which there are such numbers?"
                – Ben Voigt
                Dec 10 at 0:31












                up vote
                1
                down vote













                So it was impossible with natural numbers (see other answers)
                But if you willing to bend the rules a bit, by making that when decimal numbers are involved $A.a times B.b = AB.ab$



                then you can find



                $$x.99999999999... times 9.9999999999... = x9.99999999999...$$



                or more compactly



                $$x.(9) times 9.(9) = x9.(9)$$



                this is possible because $x.(9) = x + 1$



                and for an example if $x=4$



                $5 times 10 = 50 Leftrightarrow 4.(9) times 9.(9) = 49.(9)$






                share|cite|improve this answer

























                  up vote
                  1
                  down vote













                  So it was impossible with natural numbers (see other answers)
                  But if you willing to bend the rules a bit, by making that when decimal numbers are involved $A.a times B.b = AB.ab$



                  then you can find



                  $$x.99999999999... times 9.9999999999... = x9.99999999999...$$



                  or more compactly



                  $$x.(9) times 9.(9) = x9.(9)$$



                  this is possible because $x.(9) = x + 1$



                  and for an example if $x=4$



                  $5 times 10 = 50 Leftrightarrow 4.(9) times 9.(9) = 49.(9)$






                  share|cite|improve this answer























                    up vote
                    1
                    down vote










                    up vote
                    1
                    down vote









                    So it was impossible with natural numbers (see other answers)
                    But if you willing to bend the rules a bit, by making that when decimal numbers are involved $A.a times B.b = AB.ab$



                    then you can find



                    $$x.99999999999... times 9.9999999999... = x9.99999999999...$$



                    or more compactly



                    $$x.(9) times 9.(9) = x9.(9)$$



                    this is possible because $x.(9) = x + 1$



                    and for an example if $x=4$



                    $5 times 10 = 50 Leftrightarrow 4.(9) times 9.(9) = 49.(9)$






                    share|cite|improve this answer












                    So it was impossible with natural numbers (see other answers)
                    But if you willing to bend the rules a bit, by making that when decimal numbers are involved $A.a times B.b = AB.ab$



                    then you can find



                    $$x.99999999999... times 9.9999999999... = x9.99999999999...$$



                    or more compactly



                    $$x.(9) times 9.(9) = x9.(9)$$



                    this is possible because $x.(9) = x + 1$



                    and for an example if $x=4$



                    $5 times 10 = 50 Leftrightarrow 4.(9) times 9.(9) = 49.(9)$







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Dec 8 at 14:06









                    gota

                    404315




                    404315






















                        up vote
                        0
                        down vote













                        (1) A slightly different method:



                        Let $k>1$ be an an arbitrary base. We know that $lceillog_kBrceil$ is the number of digits in $B$ in base $k$.



                        We want to prove if there exists a solution to $A*B=A*k^{lceil log_{k}Brceil}$+B.



                        Isolate $A$ and $B$, $::1-frac{1}{A}=frac{k^{lceil log_{k}Brceil}}{B}$.
                        For positive $A$, we have that $1-frac{1}{A} < 1$.



                        Recall that by the definition of logarithm, $k^{log_kB}=B$. $:$ Since$lceil xrceilgeq x$, we know $frac{k^{lceil log_{k}Brceil}}{B}geq1$. Thus, along with the other answers, we come to a contradiction. One side of the equation is less than 1, the other at least 1.





                        (3) Now what if $k<0$?



                        I will first point out that there do exist integers $A$ and $B$ for which $A * B=AB$.



                        We will use the following definition of negative base 10:




                        A number of the form $ldots d_2d_1d_0$ with numerical value $ldots+ d_2(-10)^2+d_1(-10)^1+d_0(-10)^0$ where every $0leq d_ileq9.$




                        Now consider $A=-4_{10}=-4_{-10}$ and $B=8_{10}=8_{-10}$. Discarding the negative sign, $AB=48$.



                        $A*B=-32_{10}=4(-10)^1+8(-10)^0=48_{-10}$.



                        Another example:



                        $A=-9_{10}=-9_{-10}, B=90_{10}=90_{-10}$. $AB =990$.



                        $A*B=-810_{10}=9(-10)^1+9(-10)^1+0(-10)^0=-990_{-10}$. This time we must include the negative sign, but regardless the digits are the same.



                        So for your third question, provided that we let the negative sign in the concatenation be optional, we can find examples where multiplying two numbers will yield the concatenation of their digits.






                        share|cite|improve this answer



























                          up vote
                          0
                          down vote













                          (1) A slightly different method:



                          Let $k>1$ be an an arbitrary base. We know that $lceillog_kBrceil$ is the number of digits in $B$ in base $k$.



                          We want to prove if there exists a solution to $A*B=A*k^{lceil log_{k}Brceil}$+B.



                          Isolate $A$ and $B$, $::1-frac{1}{A}=frac{k^{lceil log_{k}Brceil}}{B}$.
                          For positive $A$, we have that $1-frac{1}{A} < 1$.



                          Recall that by the definition of logarithm, $k^{log_kB}=B$. $:$ Since$lceil xrceilgeq x$, we know $frac{k^{lceil log_{k}Brceil}}{B}geq1$. Thus, along with the other answers, we come to a contradiction. One side of the equation is less than 1, the other at least 1.





                          (3) Now what if $k<0$?



                          I will first point out that there do exist integers $A$ and $B$ for which $A * B=AB$.



                          We will use the following definition of negative base 10:




                          A number of the form $ldots d_2d_1d_0$ with numerical value $ldots+ d_2(-10)^2+d_1(-10)^1+d_0(-10)^0$ where every $0leq d_ileq9.$




                          Now consider $A=-4_{10}=-4_{-10}$ and $B=8_{10}=8_{-10}$. Discarding the negative sign, $AB=48$.



                          $A*B=-32_{10}=4(-10)^1+8(-10)^0=48_{-10}$.



                          Another example:



                          $A=-9_{10}=-9_{-10}, B=90_{10}=90_{-10}$. $AB =990$.



                          $A*B=-810_{10}=9(-10)^1+9(-10)^1+0(-10)^0=-990_{-10}$. This time we must include the negative sign, but regardless the digits are the same.



                          So for your third question, provided that we let the negative sign in the concatenation be optional, we can find examples where multiplying two numbers will yield the concatenation of their digits.






                          share|cite|improve this answer

























                            up vote
                            0
                            down vote










                            up vote
                            0
                            down vote









                            (1) A slightly different method:



                            Let $k>1$ be an an arbitrary base. We know that $lceillog_kBrceil$ is the number of digits in $B$ in base $k$.



                            We want to prove if there exists a solution to $A*B=A*k^{lceil log_{k}Brceil}$+B.



                            Isolate $A$ and $B$, $::1-frac{1}{A}=frac{k^{lceil log_{k}Brceil}}{B}$.
                            For positive $A$, we have that $1-frac{1}{A} < 1$.



                            Recall that by the definition of logarithm, $k^{log_kB}=B$. $:$ Since$lceil xrceilgeq x$, we know $frac{k^{lceil log_{k}Brceil}}{B}geq1$. Thus, along with the other answers, we come to a contradiction. One side of the equation is less than 1, the other at least 1.





                            (3) Now what if $k<0$?



                            I will first point out that there do exist integers $A$ and $B$ for which $A * B=AB$.



                            We will use the following definition of negative base 10:




                            A number of the form $ldots d_2d_1d_0$ with numerical value $ldots+ d_2(-10)^2+d_1(-10)^1+d_0(-10)^0$ where every $0leq d_ileq9.$




                            Now consider $A=-4_{10}=-4_{-10}$ and $B=8_{10}=8_{-10}$. Discarding the negative sign, $AB=48$.



                            $A*B=-32_{10}=4(-10)^1+8(-10)^0=48_{-10}$.



                            Another example:



                            $A=-9_{10}=-9_{-10}, B=90_{10}=90_{-10}$. $AB =990$.



                            $A*B=-810_{10}=9(-10)^1+9(-10)^1+0(-10)^0=-990_{-10}$. This time we must include the negative sign, but regardless the digits are the same.



                            So for your third question, provided that we let the negative sign in the concatenation be optional, we can find examples where multiplying two numbers will yield the concatenation of their digits.






                            share|cite|improve this answer














                            (1) A slightly different method:



                            Let $k>1$ be an an arbitrary base. We know that $lceillog_kBrceil$ is the number of digits in $B$ in base $k$.



                            We want to prove if there exists a solution to $A*B=A*k^{lceil log_{k}Brceil}$+B.



                            Isolate $A$ and $B$, $::1-frac{1}{A}=frac{k^{lceil log_{k}Brceil}}{B}$.
                            For positive $A$, we have that $1-frac{1}{A} < 1$.



                            Recall that by the definition of logarithm, $k^{log_kB}=B$. $:$ Since$lceil xrceilgeq x$, we know $frac{k^{lceil log_{k}Brceil}}{B}geq1$. Thus, along with the other answers, we come to a contradiction. One side of the equation is less than 1, the other at least 1.





                            (3) Now what if $k<0$?



                            I will first point out that there do exist integers $A$ and $B$ for which $A * B=AB$.



                            We will use the following definition of negative base 10:




                            A number of the form $ldots d_2d_1d_0$ with numerical value $ldots+ d_2(-10)^2+d_1(-10)^1+d_0(-10)^0$ where every $0leq d_ileq9.$




                            Now consider $A=-4_{10}=-4_{-10}$ and $B=8_{10}=8_{-10}$. Discarding the negative sign, $AB=48$.



                            $A*B=-32_{10}=4(-10)^1+8(-10)^0=48_{-10}$.



                            Another example:



                            $A=-9_{10}=-9_{-10}, B=90_{10}=90_{-10}$. $AB =990$.



                            $A*B=-810_{10}=9(-10)^1+9(-10)^1+0(-10)^0=-990_{-10}$. This time we must include the negative sign, but regardless the digits are the same.



                            So for your third question, provided that we let the negative sign in the concatenation be optional, we can find examples where multiplying two numbers will yield the concatenation of their digits.







                            share|cite|improve this answer














                            share|cite|improve this answer



                            share|cite|improve this answer








                            edited Dec 9 at 18:31

























                            answered Dec 9 at 10:47









                            bitconfused

                            2149




                            2149






















                                up vote
                                0
                                down vote













                                Here's my approach: take two natural numbers $n,m$ with $x,y$ number of digits respectively. Then in particular we can bound $$n cdot m leq (9 cdots text{($x$ times)} cdots 9) cdot (9 cdots text{($y$ times)} cdots 9) = (10^{x} - 1)(10^y - 1),$$
                                so $n cdot m leq 10^{x+y} - 10^x - 10^y + 1$. Now, we can also bound
                                $$nm geq (10 cdots text{($x-1$ zeros)} cdots 0)(10 cdots text{($y-1$ zeros)} cdots 0) geq 10 cdots text{($x+y-1$ zeros)} cdots 0,$$ so that $nm geq 10^{x+y-1}$.



                                This is as far as I've got, there are already better answers around.






                                share|cite|improve this answer























                                • Absolutely right - fixed it now. Probably would've been more standard to swap the roles of $x,y$ and $n,m$ but meh shrugs
                                  – Stuartg98
                                  Dec 9 at 21:08















                                up vote
                                0
                                down vote













                                Here's my approach: take two natural numbers $n,m$ with $x,y$ number of digits respectively. Then in particular we can bound $$n cdot m leq (9 cdots text{($x$ times)} cdots 9) cdot (9 cdots text{($y$ times)} cdots 9) = (10^{x} - 1)(10^y - 1),$$
                                so $n cdot m leq 10^{x+y} - 10^x - 10^y + 1$. Now, we can also bound
                                $$nm geq (10 cdots text{($x-1$ zeros)} cdots 0)(10 cdots text{($y-1$ zeros)} cdots 0) geq 10 cdots text{($x+y-1$ zeros)} cdots 0,$$ so that $nm geq 10^{x+y-1}$.



                                This is as far as I've got, there are already better answers around.






                                share|cite|improve this answer























                                • Absolutely right - fixed it now. Probably would've been more standard to swap the roles of $x,y$ and $n,m$ but meh shrugs
                                  – Stuartg98
                                  Dec 9 at 21:08













                                up vote
                                0
                                down vote










                                up vote
                                0
                                down vote









                                Here's my approach: take two natural numbers $n,m$ with $x,y$ number of digits respectively. Then in particular we can bound $$n cdot m leq (9 cdots text{($x$ times)} cdots 9) cdot (9 cdots text{($y$ times)} cdots 9) = (10^{x} - 1)(10^y - 1),$$
                                so $n cdot m leq 10^{x+y} - 10^x - 10^y + 1$. Now, we can also bound
                                $$nm geq (10 cdots text{($x-1$ zeros)} cdots 0)(10 cdots text{($y-1$ zeros)} cdots 0) geq 10 cdots text{($x+y-1$ zeros)} cdots 0,$$ so that $nm geq 10^{x+y-1}$.



                                This is as far as I've got, there are already better answers around.






                                share|cite|improve this answer














                                Here's my approach: take two natural numbers $n,m$ with $x,y$ number of digits respectively. Then in particular we can bound $$n cdot m leq (9 cdots text{($x$ times)} cdots 9) cdot (9 cdots text{($y$ times)} cdots 9) = (10^{x} - 1)(10^y - 1),$$
                                so $n cdot m leq 10^{x+y} - 10^x - 10^y + 1$. Now, we can also bound
                                $$nm geq (10 cdots text{($x-1$ zeros)} cdots 0)(10 cdots text{($y-1$ zeros)} cdots 0) geq 10 cdots text{($x+y-1$ zeros)} cdots 0,$$ so that $nm geq 10^{x+y-1}$.



                                This is as far as I've got, there are already better answers around.







                                share|cite|improve this answer














                                share|cite|improve this answer



                                share|cite|improve this answer








                                edited Dec 9 at 21:07

























                                answered Dec 8 at 0:58









                                Stuartg98

                                586




                                586












                                • Absolutely right - fixed it now. Probably would've been more standard to swap the roles of $x,y$ and $n,m$ but meh shrugs
                                  – Stuartg98
                                  Dec 9 at 21:08


















                                • Absolutely right - fixed it now. Probably would've been more standard to swap the roles of $x,y$ and $n,m$ but meh shrugs
                                  – Stuartg98
                                  Dec 9 at 21:08
















                                Absolutely right - fixed it now. Probably would've been more standard to swap the roles of $x,y$ and $n,m$ but meh shrugs
                                – Stuartg98
                                Dec 9 at 21:08




                                Absolutely right - fixed it now. Probably would've been more standard to swap the roles of $x,y$ and $n,m$ but meh shrugs
                                – Stuartg98
                                Dec 9 at 21:08


















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