What is the greatest common divisor of the numbers $9 m+7 n$ and $3 m+2 n$, if the numbers $m$ and $n$ have no common divisors other than one?

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Any common divisor of the numbers $9 m+7 n$ and $3 m+2 n$ must also be a divisor of the numbers $(9 m+7 n)-3(3 m+2 n) = n$ and $7(3 m+2 n)-2(9 m+7 n) = 3 m$. Since the numbers $m$ and $n$ are coprime, any common divisor of the numbers $n$ and $3 m$ must be a divisor of the number 3, that is, it cannot be greater than 3.

This value is possible: for $m=1, n=3$ GCD $(9 m+7 n, 3 m+2 n)=$ GCD(30,9$)=3$.

## Answer

3.