Modulus with divisor unknown












0














I'm trying to find the answer to this question, but more importantly, a general formula.




$x$ is a positive integer which solves the following questions: when $280$ is divided by $x$, the remainder is $16$; when $900$ is divided by $x$, the remainder is $42$. Find the greatest possible value of $x$.




It's quite obvious that $43leq xleq264$, and I've converted the above into the equations $280=kx+16$ and $900=mx+42$. I'm not sure how to proceed further.






algebra-precalculus modular-arithmetic







share|cite|improve this question






















  • Is my answer right? Please check once .
    – Akash Roy
    Nov 27 at 14:07
















0














I'm trying to find the answer to this question, but more importantly, a general formula.




$x$ is a positive integer which solves the following questions: when $280$ is divided by $x$, the remainder is $16$; when $900$ is divided by $x$, the remainder is $42$. Find the greatest possible value of $x$.




It's quite obvious that $43leq xleq264$, and I've converted the above into the equations $280=kx+16$ and $900=mx+42$. I'm not sure how to proceed further.






algebra-precalculus modular-arithmetic







share|cite|improve this question






















  • Is my answer right? Please check once .
    – Akash Roy
    Nov 27 at 14:07














0












0








0







I'm trying to find the answer to this question, but more importantly, a general formula.




$x$ is a positive integer which solves the following questions: when $280$ is divided by $x$, the remainder is $16$; when $900$ is divided by $x$, the remainder is $42$. Find the greatest possible value of $x$.




It's quite obvious that $43leq xleq264$, and I've converted the above into the equations $280=kx+16$ and $900=mx+42$. I'm not sure how to proceed further.






algebra-precalculus modular-arithmetic







share|cite|improve this question













I'm trying to find the answer to this question, but more importantly, a general formula.




$x$ is a positive integer which solves the following questions: when $280$ is divided by $x$, the remainder is $16$; when $900$ is divided by $x$, the remainder is $42$. Find the greatest possible value of $x$.




It's quite obvious that $43leq xleq264$, and I've converted the above into the equations $280=kx+16$ and $900=mx+42$. I'm not sure how to proceed further.






algebra-precalculus modular-arithmetic




algebra-precalculus modular-arithmetic






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 27 at 13:07









Kyky

444213




444213












  • Is my answer right? Please check once .
    – Akash Roy
    Nov 27 at 14:07


















  • Is my answer right? Please check once .
    – Akash Roy
    Nov 27 at 14:07
















Is my answer right? Please check once .
– Akash Roy
Nov 27 at 14:07




Is my answer right? Please check once .
– Akash Roy
Nov 27 at 14:07










3 Answers
3






active

oldest

votes


















3














From $280 = kx + 16$ and $900 = mx + 42$, simple transpositions give us $858 = mx$ and $264 = kx$. Thus, $x$ is a divisor of $858$ and $264$. What is the greatest such divisor?(Think : greatest common divisor). Do you know a "general" procedure for finding the greatest common divisor?



Do you think that this satisfies the conditions of the question, and is the largest such number to do so?






share|cite|improve this answer





















  • Thanks! I'm surprised I haven't come up with the answer myself now that I think of it.
    – Kyky
    Nov 27 at 13:14










  • Actually, I should point out : suppose this gcd were less than $42$, then the remainder upon division cannot be $42$. So we are restricted to divisors of $264$ which are greater than $42$, and there are not too many to check, so in this case one need not even go through the Euclidean algorithm.
    – астон вілла олоф мэллбэрг
    Nov 27 at 13:19



















0














Hint:



When $280$ is divided by $x$, the remainder is $16$. What does this tell you about the remainder when $280 - 16$ is divided by $x$?






share|cite|improve this answer





















  • Is my answer right?
    – Akash Roy
    Nov 27 at 14:08



















0














The number which you are looking for has to be the HCF of $858$ and $264$ because the two conditions have to be fulfilled simultaneously. Thus the answer is $66$. You can use Euclid Algorithm or Prime factorization method to find HCF.



Note : HCF denotes Highest Common Factor






share|cite|improve this answer























  • HCF? What is HCF?
    – 5xum
    Nov 27 at 17:06










  • Highest common factor @5xum or Greatest common divisor
    – Akash Roy
    Nov 27 at 17:07










  • Is the answer right?
    – Akash Roy
    Nov 27 at 17:08










  • It's correct, but I think we should spare the extra 1.5 seconds to write out ambiguous abreviations such as HCF...
    – 5xum
    Nov 27 at 17:30










  • Should I replace it with GCD?
    – Akash Roy
    Nov 27 at 17:31











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3 Answers
3






active

oldest

votes








3 Answers
3






active

oldest

votes









active

oldest

votes






active

oldest

votes









3














From $280 = kx + 16$ and $900 = mx + 42$, simple transpositions give us $858 = mx$ and $264 = kx$. Thus, $x$ is a divisor of $858$ and $264$. What is the greatest such divisor?(Think : greatest common divisor). Do you know a "general" procedure for finding the greatest common divisor?



Do you think that this satisfies the conditions of the question, and is the largest such number to do so?






share|cite|improve this answer





















  • Thanks! I'm surprised I haven't come up with the answer myself now that I think of it.
    – Kyky
    Nov 27 at 13:14










  • Actually, I should point out : suppose this gcd were less than $42$, then the remainder upon division cannot be $42$. So we are restricted to divisors of $264$ which are greater than $42$, and there are not too many to check, so in this case one need not even go through the Euclidean algorithm.
    – астон вілла олоф мэллбэрг
    Nov 27 at 13:19
















3














From $280 = kx + 16$ and $900 = mx + 42$, simple transpositions give us $858 = mx$ and $264 = kx$. Thus, $x$ is a divisor of $858$ and $264$. What is the greatest such divisor?(Think : greatest common divisor). Do you know a "general" procedure for finding the greatest common divisor?



Do you think that this satisfies the conditions of the question, and is the largest such number to do so?






share|cite|improve this answer





















  • Thanks! I'm surprised I haven't come up with the answer myself now that I think of it.
    – Kyky
    Nov 27 at 13:14










  • Actually, I should point out : suppose this gcd were less than $42$, then the remainder upon division cannot be $42$. So we are restricted to divisors of $264$ which are greater than $42$, and there are not too many to check, so in this case one need not even go through the Euclidean algorithm.
    – астон вілла олоф мэллбэрг
    Nov 27 at 13:19














3












3








3






From $280 = kx + 16$ and $900 = mx + 42$, simple transpositions give us $858 = mx$ and $264 = kx$. Thus, $x$ is a divisor of $858$ and $264$. What is the greatest such divisor?(Think : greatest common divisor). Do you know a "general" procedure for finding the greatest common divisor?



Do you think that this satisfies the conditions of the question, and is the largest such number to do so?






share|cite|improve this answer












From $280 = kx + 16$ and $900 = mx + 42$, simple transpositions give us $858 = mx$ and $264 = kx$. Thus, $x$ is a divisor of $858$ and $264$. What is the greatest such divisor?(Think : greatest common divisor). Do you know a "general" procedure for finding the greatest common divisor?



Do you think that this satisfies the conditions of the question, and is the largest such number to do so?







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 27 at 13:11









астон вілла олоф мэллбэрг

37.2k33376




37.2k33376












  • Thanks! I'm surprised I haven't come up with the answer myself now that I think of it.
    – Kyky
    Nov 27 at 13:14










  • Actually, I should point out : suppose this gcd were less than $42$, then the remainder upon division cannot be $42$. So we are restricted to divisors of $264$ which are greater than $42$, and there are not too many to check, so in this case one need not even go through the Euclidean algorithm.
    – астон вілла олоф мэллбэрг
    Nov 27 at 13:19


















  • Thanks! I'm surprised I haven't come up with the answer myself now that I think of it.
    – Kyky
    Nov 27 at 13:14










  • Actually, I should point out : suppose this gcd were less than $42$, then the remainder upon division cannot be $42$. So we are restricted to divisors of $264$ which are greater than $42$, and there are not too many to check, so in this case one need not even go through the Euclidean algorithm.
    – астон вілла олоф мэллбэрг
    Nov 27 at 13:19
















Thanks! I'm surprised I haven't come up with the answer myself now that I think of it.
– Kyky
Nov 27 at 13:14




Thanks! I'm surprised I haven't come up with the answer myself now that I think of it.
– Kyky
Nov 27 at 13:14












Actually, I should point out : suppose this gcd were less than $42$, then the remainder upon division cannot be $42$. So we are restricted to divisors of $264$ which are greater than $42$, and there are not too many to check, so in this case one need not even go through the Euclidean algorithm.
– астон вілла олоф мэллбэрг
Nov 27 at 13:19




Actually, I should point out : suppose this gcd were less than $42$, then the remainder upon division cannot be $42$. So we are restricted to divisors of $264$ which are greater than $42$, and there are not too many to check, so in this case one need not even go through the Euclidean algorithm.
– астон вілла олоф мэллбэрг
Nov 27 at 13:19











0














Hint:



When $280$ is divided by $x$, the remainder is $16$. What does this tell you about the remainder when $280 - 16$ is divided by $x$?






share|cite|improve this answer





















  • Is my answer right?
    – Akash Roy
    Nov 27 at 14:08
















0














Hint:



When $280$ is divided by $x$, the remainder is $16$. What does this tell you about the remainder when $280 - 16$ is divided by $x$?






share|cite|improve this answer





















  • Is my answer right?
    – Akash Roy
    Nov 27 at 14:08














0












0








0






Hint:



When $280$ is divided by $x$, the remainder is $16$. What does this tell you about the remainder when $280 - 16$ is divided by $x$?






share|cite|improve this answer












Hint:



When $280$ is divided by $x$, the remainder is $16$. What does this tell you about the remainder when $280 - 16$ is divided by $x$?







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 27 at 13:11









5xum

89.5k393161




89.5k393161












  • Is my answer right?
    – Akash Roy
    Nov 27 at 14:08


















  • Is my answer right?
    – Akash Roy
    Nov 27 at 14:08
















Is my answer right?
– Akash Roy
Nov 27 at 14:08




Is my answer right?
– Akash Roy
Nov 27 at 14:08











0














The number which you are looking for has to be the HCF of $858$ and $264$ because the two conditions have to be fulfilled simultaneously. Thus the answer is $66$. You can use Euclid Algorithm or Prime factorization method to find HCF.



Note : HCF denotes Highest Common Factor






share|cite|improve this answer























  • HCF? What is HCF?
    – 5xum
    Nov 27 at 17:06










  • Highest common factor @5xum or Greatest common divisor
    – Akash Roy
    Nov 27 at 17:07










  • Is the answer right?
    – Akash Roy
    Nov 27 at 17:08










  • It's correct, but I think we should spare the extra 1.5 seconds to write out ambiguous abreviations such as HCF...
    – 5xum
    Nov 27 at 17:30










  • Should I replace it with GCD?
    – Akash Roy
    Nov 27 at 17:31
















0














The number which you are looking for has to be the HCF of $858$ and $264$ because the two conditions have to be fulfilled simultaneously. Thus the answer is $66$. You can use Euclid Algorithm or Prime factorization method to find HCF.



Note : HCF denotes Highest Common Factor






share|cite|improve this answer























  • HCF? What is HCF?
    – 5xum
    Nov 27 at 17:06










  • Highest common factor @5xum or Greatest common divisor
    – Akash Roy
    Nov 27 at 17:07










  • Is the answer right?
    – Akash Roy
    Nov 27 at 17:08










  • It's correct, but I think we should spare the extra 1.5 seconds to write out ambiguous abreviations such as HCF...
    – 5xum
    Nov 27 at 17:30










  • Should I replace it with GCD?
    – Akash Roy
    Nov 27 at 17:31














0












0








0






The number which you are looking for has to be the HCF of $858$ and $264$ because the two conditions have to be fulfilled simultaneously. Thus the answer is $66$. You can use Euclid Algorithm or Prime factorization method to find HCF.



Note : HCF denotes Highest Common Factor






share|cite|improve this answer














The number which you are looking for has to be the HCF of $858$ and $264$ because the two conditions have to be fulfilled simultaneously. Thus the answer is $66$. You can use Euclid Algorithm or Prime factorization method to find HCF.



Note : HCF denotes Highest Common Factor







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Nov 27 at 17:08

























answered Nov 27 at 13:35









Akash Roy

1




1












  • HCF? What is HCF?
    – 5xum
    Nov 27 at 17:06










  • Highest common factor @5xum or Greatest common divisor
    – Akash Roy
    Nov 27 at 17:07










  • Is the answer right?
    – Akash Roy
    Nov 27 at 17:08










  • It's correct, but I think we should spare the extra 1.5 seconds to write out ambiguous abreviations such as HCF...
    – 5xum
    Nov 27 at 17:30










  • Should I replace it with GCD?
    – Akash Roy
    Nov 27 at 17:31


















  • HCF? What is HCF?
    – 5xum
    Nov 27 at 17:06










  • Highest common factor @5xum or Greatest common divisor
    – Akash Roy
    Nov 27 at 17:07










  • Is the answer right?
    – Akash Roy
    Nov 27 at 17:08










  • It's correct, but I think we should spare the extra 1.5 seconds to write out ambiguous abreviations such as HCF...
    – 5xum
    Nov 27 at 17:30










  • Should I replace it with GCD?
    – Akash Roy
    Nov 27 at 17:31
















HCF? What is HCF?
– 5xum
Nov 27 at 17:06




HCF? What is HCF?
– 5xum
Nov 27 at 17:06












Highest common factor @5xum or Greatest common divisor
– Akash Roy
Nov 27 at 17:07




Highest common factor @5xum or Greatest common divisor
– Akash Roy
Nov 27 at 17:07












Is the answer right?
– Akash Roy
Nov 27 at 17:08




Is the answer right?
– Akash Roy
Nov 27 at 17:08












It's correct, but I think we should spare the extra 1.5 seconds to write out ambiguous abreviations such as HCF...
– 5xum
Nov 27 at 17:30




It's correct, but I think we should spare the extra 1.5 seconds to write out ambiguous abreviations such as HCF...
– 5xum
Nov 27 at 17:30












Should I replace it with GCD?
– Akash Roy
Nov 27 at 17:31




Should I replace it with GCD?
– Akash Roy
Nov 27 at 17:31


















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