Question

Find HCF of 65 and 117 and find a pair of integral values of m and n such that HCF =65m + 117n.

Answer 1

By Euclid's division algorithm

$117=65\times 1+52\to \left(1\right)65=52\times 1+13\to \left(2\right)52=13\times 4+0\to \left(3\right)$

Therefore $13$ is HCF f $(65,117)$

Now going backward

$13=65+52\times (-1)13=65+[117-65\times 1]\times (-1)$

[From $\left(1\right)$]

$13=65\times 2+117\times (-1)\therefore m=2n=-1$