Question

Use division algorithm to show that any positive odd integer is of the form $6q+1$ or $6q+3$ or $6q+5$, where $q$ is some integer.

Answer 1

Let $a$ be any positive integer and $b=6.$

Then by division algorithm,

$a=6q+r$ where $r=0,1,2,3,4,5$

So, $a$ is of the form $6q,6q+1,6q+2,6q+3,6q+3,6q+4,6q+5.$

Therefore if $a$ is an odd integer then $a$ is of the form $6q+1$ or $6q+3$ $6q+5.$

Hence, any positive odd integer is of the form $6q+1$ or $6q+3$ or $6q+5.$