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You could also use the fact that (A_1 A_4)^2 + (A_4 A_7)^2 = (A_1 A_7)^2 (pythagorean theorem) and same when it is A_2 instead of A_4, etc.
I think you forgot the "acute" part.
If c was the longest side, then c<17 (because 8-15-17 is a Pythagorean Triple).
If c was one of the shorter sides, then it has to be 8^2+c^2>15^2
so 64+c^2>225
so c^2>161
so c>sqrt(161)
sqrt(161)<c<17
13, 14, 15, and 16 work.
(I might be wrong)
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