Try to prove that!

Given that if $N$ is divided by $m,$ the remainder is $7.$

Hence $N = mx + 7$ where $x$ is an integer.

So $5N = 5mx + 35.$

On the other hand, $5N = my + 10$ where $y$ is an integer.

Hence $5mx + 35 = my + 10$ which implies $m(y - 5x) = 25.$

Thus $25$ is divisible by $m.$

Since the only divisors of $25$ are $1, 5$ and $25,$ $m$ may only be one of these three numbers.

But we know that if a number is divided by another number, the remainder is less than the divisor.

Therefore $m$ can be neither $1$ nor $5$ as the two remainders obtained are $7$ and $10.$

Hence $m = \boxed{25}$