Question

Prove that if a positive integer is of the form $6q+5$, then it is of the form $3q+2$ for some integer $q$, but not conversely.

Answer 1

Let $n=6q+5,$ where $q$ is a positive integer.

We know that any positive integer is of the form$q=3kor,3k+1or,3k+2$

If $q=3k,$ then

$n=6q+5=18k+5=3(6k+1)+2=3m+2,$ where $m=6k+1$

If $q=3k+1,$ then

$n=6q+5=6(3k+1)+5=3(6k+3)+2=3m+2,$ where $m=6k+3$

If $q=3k+2,$ then

$n=6q+5=6(3k+2)+5=3(6k+5)+2=3m+2,$ where $m=6k+5.$