Finding smallest a+b from given conditions. I'm stuck on a decade old olympiad problem which is as follow:
Find least possible value of a+b where a,b are positive integers such that 11 divides a+13b and 13 divides a+11b.
I m clueless. I tried something like a+11b=13k and a+13b=11m (for some positive integer k and m), but that didn't work.
Hoping for help,I shall be thankful if you give an easy answer with a bit of explaination.

Hint
we have
$a+13b \equiv 0 \mod 11$,
$a+11b \equiv 0 \mod 13$
$\implies$
$a+2b \equiv 0 \mod 11$,
$a-2b \equiv 0 \mod 13$
$\implies$
$2a =11p + 13q$ and $4b=11p-13q$.
thus
$a=23$ and $b=5$ with $p=3,q=1$.
