# Finding smallest a+b from given condintions.

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 $p=3,q=1$.

• no it is correct,I have edited it(there was no comma) – Vidyanshu Mishra Nov 13 '16 at 8:25
• You're right, sorry – Ricardo Largaespada Nov 13 '16 at 8:28