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# GCD stuff

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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
+1

m = 24

n = 36

GCD =[24, 36] = 12

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

Jul 31, 2019
#2
+25648
+2

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

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

Jul 31, 2019