smc

Let the number of students be $n\ge 5$. Then the sum of their scores is at least $5\cdot 100+(n-5)\cdot 60$. At the same time, we need to achieve the mean $76$, which is equivalent to achieving the sum $76n$. Hence we get a necessary condition on $n$: we must have $5\cdot 100+(n-5)\cdot 60\le 76n$. This can be simplified to $200\le 16n$. The smallest integer $n$ for which this is true is $n=13$. To finish our solution, we now need to find one way how $13$ students could have scored on the test. We have $13\cdot 76=988$ points to divide among them. The five $100$s make $500$, hence we must divide the remaining $488$ points among the other $8$ students. This can be done e.g. by giving $61$ points to each of them. Hence the smallest possible number of students is $\boxed{13}$.