Indices mistakes-3

We know $a^{mn}=(a^m)(a^m)(a^m)...(a^m)\text{n times}$ But there are some students who claim that $a^{mn}=(a^m)(a^n)$ Which is obviously incorrect. But not really, since it is true for some triplets of integers, $a,m,n$ . If $a\in[1,10]$ and $m,n\in[0,10]$ , find the number of possible triplets.