Value of a+b+c

There are positive integers a,b, and c that satisfy this system of two equations:
c^2-a^2-b^2=101
ab=72
What is the value of a+b+c?
Source: NCTM Mathematics Teacher, November 2006

SOLUTION
Multiply the first equation by \textrm{-}1, the second equation by 2 and add them
\textrm{-}c^2+a^2+b^2=\textrm{-}101
2ab=144
\overline{a^2+2ab+b^2-c^2=43}
(a+b)^2-c^2=43
(a+b+c)(a+b-c)=43
43 =1\times 43 because 43 is a prime number.
a+b+c cannot equal 1 because a,b,c are positive integers.
a+b+c=43 and a+b-c=1.

Answer: 43

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About mvtrinh

Retired high school math teacher.
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