SOLUTION: a rectangle has length a, width b and diagonal c. All three are integers less than 100, c is 40 units less than the sum of a and b, and they have no common factors. find a, b, c

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Question 262758: a rectangle has length a, width b and diagonal c. All three are integers less than 100, c is 40 units less than the sum of a and b, and they have no common factors. find a, b, c
Answer by drk(1908)   (Show Source): You can put this solution on YOUR website!
FRom the given, we know that

we also now that c= a+b - 40
By substitution, we get

expanding the right side, we get

subtracting a^2 and b^2 from both sides, we get

or

we can solve for a in terms of b. factor out 2a to get

subtract 80b to get

divide by 2(40-b) to get

which simplifies to

At this point we know that b <= 20 OR b > 40.
Since we want integers for a and b, here are a few options in (a,b) form:
(8,15) or (15,8)
If we use
(8,15), then c = 17 and all 3 share no common factors. But this doesn't satisfy our restrictions.
Our only new options for (a,b,c) are:
(67, 72, 97)
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