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
+26367 If m and n are positive integers such that \(gcd(m,n)=12\) , then what is the smallest possible value of \(gcd(10m,15n)\) ?
\(\begin{array}{|rclcrcll|} \hline m&=&12 && n &=& 12 \quad &| \quad \text{$m=12$ and $n=12$ are the smallest values for $m$ and $n$}\\ && && && \quad &| \quad \text{to get $gcd(12,12)=12$ } \\ &=&2^23 && &=&2^23=12 \quad & | \quad \text{after factorization} \\ && \gcd(12,12) &=& 2^23=12 && \quad & | \quad \text{multiply all prim numbers with the lowest exponent } \\\\ m&=&10*12 && n &=& 15*12 \\ &=&2^33*5 && &=&2^23^25 \quad & | \quad \text{after factorization} \\ && \gcd(120,180) &=& 2^23*5=\mathbf{60} && \quad & | \quad \text{multiply all prim numbers with the lowest exponent } \\ \hline \end{array}\)
