number theory lcm [duplicate]
This question already has an answer here:
LCM is associative: $ text{lcm}(text{lcm}(a,b),c)=text{lcm}(a,text{lcm}(b,c))$
2 answers
prove if $a, b ,c$ are positive integers, then $lcm(a,lcm(b,c))=lcm(a,b,c)=lcm(lcm(a,b),c)$
$lcm$ is least common multiple
My thought is to show that they have common divisors but not sure how to go about it.
elementary-number-theory
asked Nov 27 at 6:45
cap
103
marked as duplicate by Bill Dubuque elementary-number-theory
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Dec 1 at 18:43
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
This question already has an answer here:
LCM is associative: $ text{lcm}(text{lcm}(a,b),c)=text{lcm}(a,text{lcm}(b,c))$
2 answers
prove if $a, b ,c$ are positive integers, then $lcm(a,lcm(b,c))=lcm(a,b,c)=lcm(lcm(a,b),c)$
$lcm$ is least common multiple
My thought is to show that they have common divisors but not sure how to go about it.
elementary-number-theory
asked Nov 27 at 6:45
cap
103
marked as duplicate by Bill Dubuque elementary-number-theory
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Dec 1 at 18:43
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Observe that any common multiple of $a$ and $b$ is a multiple of $text{lcm}(a,b)$.
– Lord Shark the Unknown
Nov 27 at 6:54
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Nov 27 at 6:58
math.stackexchange.com/questions/254704/…
– 1ENİGMA1
Nov 27 at 7:33
add a comment |
This question already has an answer here:
LCM is associative: $ text{lcm}(text{lcm}(a,b),c)=text{lcm}(a,text{lcm}(b,c))$
2 answers
prove if $a, b ,c$ are positive integers, then $lcm(a,lcm(b,c))=lcm(a,b,c)=lcm(lcm(a,b),c)$
$lcm$ is least common multiple
My thought is to show that they have common divisors but not sure how to go about it.
elementary-number-theory
asked Nov 27 at 6:45
cap
103
This question already has an answer here:
LCM is associative: $ text{lcm}(text{lcm}(a,b),c)=text{lcm}(a,text{lcm}(b,c))$
2 answers
prove if $a, b ,c$ are positive integers, then $lcm(a,lcm(b,c))=lcm(a,b,c)=lcm(lcm(a,b),c)$
$lcm$ is least common multiple
My thought is to show that they have common divisors but not sure how to go about it.
This question already has an answer here:
LCM is associative: $ text{lcm}(text{lcm}(a,b),c)=text{lcm}(a,text{lcm}(b,c))$
2 answers
elementary-number-theory
elementary-number-theory
asked Nov 27 at 6:45
cap
103
asked Nov 27 at 6:45
cap
103
asked Nov 27 at 6:45
cap
103
asked Nov 27 at 6:45
cap
103
asked Nov 27 at 6:45
cap
103
103
marked as duplicate by Bill Dubuque elementary-number-theory
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Dec 1 at 18:43
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marked as duplicate by Bill Dubuque elementary-number-theory
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Dec 1 at 18:43
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Observe that any common multiple of $a$ and $b$ is a multiple of $text{lcm}(a,b)$.
– Lord Shark the Unknown
Nov 27 at 6:54
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Nov 27 at 6:58
math.stackexchange.com/questions/254704/…
– 1ENİGMA1
Nov 27 at 7:33
add a comment |
Observe that any common multiple of $a$ and $b$ is a multiple of $text{lcm}(a,b)$.
– Lord Shark the Unknown
Nov 27 at 6:54
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Nov 27 at 6:58
math.stackexchange.com/questions/254704/…
– 1ENİGMA1
Nov 27 at 7:33
Observe that any common multiple of $a$ and $b$ is a multiple of $text{lcm}(a,b)$.
– Lord Shark the Unknown
Nov 27 at 6:54
Observe that any common multiple of $a$ and $b$ is a multiple of $text{lcm}(a,b)$.
– Lord Shark the Unknown
Nov 27 at 6:54
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Nov 27 at 6:58
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Nov 27 at 6:58
math.stackexchange.com/questions/254704/…
– 1ENİGMA1
Nov 27 at 7:33
math.stackexchange.com/questions/254704/…
– 1ENİGMA1
Nov 27 at 7:33
add a comment |
1 Answer
1
active
oldest
votes
Hint:
If the highest exponents of prime $p$ in $a,b,c$ be $A,B,C$ respectively.
WLOG $Age Bge C$
Now the highest exponents of prime $p$ in lcm$(a,b,c)$ will be max$(A,B,C)=A$
lcm$(a,$lcm$(b,c))=$max$(A,$max$(B,C))=$max$(A,B)=A$
lcm$($lcm$(a,b),c))=$max$(A,C)=A$
This holds true any prime that divides at least one of $a,b,c$
answered Nov 27 at 8:57
lab bhattacharjee
223k15156274
Would finding the greatest common multiple be the same proof?
– cap
Nov 27 at 17:22
@Lord, definitely
– lab bhattacharjee
Nov 27 at 17:40
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint:
If the highest exponents of prime $p$ in $a,b,c$ be $A,B,C$ respectively.
WLOG $Age Bge C$
Now the highest exponents of prime $p$ in lcm$(a,b,c)$ will be max$(A,B,C)=A$
lcm$(a,$lcm$(b,c))=$max$(A,$max$(B,C))=$max$(A,B)=A$
lcm$($lcm$(a,b),c))=$max$(A,C)=A$
This holds true any prime that divides at least one of $a,b,c$
answered Nov 27 at 8:57
lab bhattacharjee
223k15156274
Would finding the greatest common multiple be the same proof?
– cap
Nov 27 at 17:22
@Lord, definitely
– lab bhattacharjee
Nov 27 at 17:40
add a comment |
Hint:
If the highest exponents of prime $p$ in $a,b,c$ be $A,B,C$ respectively.
WLOG $Age Bge C$
Now the highest exponents of prime $p$ in lcm$(a,b,c)$ will be max$(A,B,C)=A$
lcm$(a,$lcm$(b,c))=$max$(A,$max$(B,C))=$max$(A,B)=A$
lcm$($lcm$(a,b),c))=$max$(A,C)=A$
This holds true any prime that divides at least one of $a,b,c$
answered Nov 27 at 8:57
lab bhattacharjee
223k15156274
Would finding the greatest common multiple be the same proof?
– cap
Nov 27 at 17:22
@Lord, definitely
– lab bhattacharjee
Nov 27 at 17:40
add a comment |
Hint:
If the highest exponents of prime $p$ in $a,b,c$ be $A,B,C$ respectively.
WLOG $Age Bge C$
Now the highest exponents of prime $p$ in lcm$(a,b,c)$ will be max$(A,B,C)=A$
lcm$(a,$lcm$(b,c))=$max$(A,$max$(B,C))=$max$(A,B)=A$
lcm$($lcm$(a,b),c))=$max$(A,C)=A$
This holds true any prime that divides at least one of $a,b,c$
answered Nov 27 at 8:57
lab bhattacharjee
223k15156274
Hint:
If the highest exponents of prime $p$ in $a,b,c$ be $A,B,C$ respectively.
WLOG $Age Bge C$
Now the highest exponents of prime $p$ in lcm$(a,b,c)$ will be max$(A,B,C)=A$
lcm$(a,$lcm$(b,c))=$max$(A,$max$(B,C))=$max$(A,B)=A$
lcm$($lcm$(a,b),c))=$max$(A,C)=A$
This holds true any prime that divides at least one of $a,b,c$
answered Nov 27 at 8:57
lab bhattacharjee
223k15156274
answered Nov 27 at 8:57
lab bhattacharjee
223k15156274
answered Nov 27 at 8:57
lab bhattacharjee
223k15156274
answered Nov 27 at 8:57
lab bhattacharjee
223k15156274
223k15156274
Would finding the greatest common multiple be the same proof?
– cap
Nov 27 at 17:22
@Lord, definitely
– lab bhattacharjee
Nov 27 at 17:40
add a comment |
Would finding the greatest common multiple be the same proof?
– cap
Nov 27 at 17:22
@Lord, definitely
– lab bhattacharjee
Nov 27 at 17:40
Would finding the greatest common multiple be the same proof?
– cap
Nov 27 at 17:22
Would finding the greatest common multiple be the same proof?
– cap
Nov 27 at 17:22
@Lord, definitely
– lab bhattacharjee
Nov 27 at 17:40
@Lord, definitely
– lab bhattacharjee
Nov 27 at 17:40
add a comment |
Observe that any common multiple of $a$ and $b$ is a multiple of $text{lcm}(a,b)$.
– Lord Shark the Unknown
Nov 27 at 6:54
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Nov 27 at 6:58
math.stackexchange.com/questions/254704/…
– 1ENİGMA1
Nov 27 at 7:33