## 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: $43$