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If m and n are positive integers such that gcd(m,n)=12 , then what is the smallest possible value of gcd(10m,15n) ?

 

Edit: ans=60

 Jul 31, 2019
edited by Guest  Jul 31, 2019
edited by Guest  Jul 31, 2019
edited by Guest  Jul 31, 2019
 #1
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+1

m = 24

n = 36

GCD =[24, 36] = 12

GCD[24 *10, 36 * 15] =GCD[240, 540] =60

 Jul 31, 2019
 #2
avatar+23850 
+2

If m and n are positive integers such that \(gcd(m,n)=12\) , then what is the smallest possible value of \(gcd(10m,15n)\) ?

 

\(\begin{array}{|rclcrcll|} \hline m&=&12 && n &=& 12 \quad &| \quad \text{$m=12$ and $n=12$ are the smallest values for $m$ and $n$}\\ && && && \quad &| \quad \text{to get $gcd(12,12)=12$ } \\ &=&2^23 && &=&2^23=12 \quad & | \quad \text{after factorization} \\ && \gcd(12,12) &=& 2^23=12 && \quad & | \quad \text{multiply all prim numbers with the lowest exponent } \\\\ m&=&10*12 && n &=& 15*12 \\ &=&2^33*5 && &=&2^23^25 \quad & | \quad \text{after factorization} \\ && \gcd(120,180) &=& 2^23*5=\mathbf{60} && \quad & | \quad \text{multiply all prim numbers with the lowest exponent } \\ \hline \end{array}\)

 

laugh

 Jul 31, 2019

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