Notice a few things here:

- If her average after n tests is always an integer, then after n tests, the sum of her scores will always be divisible by n.
- So after six tests, the sum of her score is a multiple of 6. Let's call that sum, after six tests, "S"
- If her scores are between 91 and 100 every time, and her scores are always different, S is definitely between 6*93 and 6*98. Since S is also a multiple of 6, it can only be 6*94, 6*95, 6*96 or 6*97
- Finally, when we add her seventh test score, 95 to S, we must get a multiple of 7. The remainder is 4 when we divide 95 by 7, so when we divide S by 7, the remainder will need to be 3, if S+95 will give us a multiple of 7.

So we just need to work out which of the few possible values of S, 6*94, 6*95, 6*96 or 6*97, gives us a remainder of 3 when we divide it by 7. Using remainder arithmetic, or inspection, you can find that S = 6*95 = 570.

So after six tests, her scores add up to 570. When we subtract from that her sixth test score, we need to get a multiple of 5. But 570 is already a multiple of 5, so her sixth test score must be a multiple of 5 too. It can't be 95, because that's her sixth score, and she never repeated scores. The only other possibility between 91 and 100 inclusive is 100, so that must be the answer.