## Least Right Side

The sides of a right triangle are all positive integers. Two of the sides are odd numbers that differ by 50. What is the least possible value for the third side?
Source: mathcontest.olemiss.edu 2/18/2013

SOLUTION

By the Pythagorean theorem
$a^2+b^2=c^2$
$a^2=c^2-b^2$
$=\left (c+b\right )\left (c-b\right )$
$=\left (c+b\right )\left (50\right )$
The least perfect square number that is divisible by $50$ is $3600$
$a^2=3600$
$a=60$
Verification
$3600=\left (c+b\right )50$
$c+b=72$
$c-b=50$
$2c=122$
$c=61$
$b=61-50=11$
$60^2+11^2=61^2$

Answer: $60$