Question

Find least positive value of $a+b$ where $a,b$ are positive integers such that $11|a+13band13|a+11b$

Answer 1

We have $13|a+11b$

$\Rightarrow 13|a-2b$ and hence $13|6a-12b$ this implies $13|6a+b$

Similarly,

$11|a+13b\Rightarrow 11|a+2b\Rightarrow 11|6a+12b\Rightarrow 11|6a+b$

Since $\text{G.C.D}(11,13)=1$

We conclude $143|6a+b$

Thus we may write $6a+b=143k$ for some integer $k$

Hence, $6a+6b=143k+5b=144k+6b-(k+b)$

This shows that $6|k+b$ and hence $k+b\ge 6$

We therefore obtain

$6(a+b)=143k+5b=138k+5(k+b)\ge 138+(5\times 6)=168$

It follows that $a+b\ge 28$

Taking $a=23$ and $b=5$

we see that the conditions of the problem satisfied. Thus the minimum value of $a+b$ is $28$