Question

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 a given integer.

On dividing a by 6 , we get $q$ as the quotient and $r$ as the remainder such that

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

when $r=0$

$a=6q$ ,even no

when $r=1$

$a=6q+1$, odd no

when$r=2$

$a=6q+2$, even no

when$r=3$

$a=6q+3$,odd no

when $r=4$

$a=6q+4$,even no

when $r=5,$

$a=6q+5$ , odd no

Any positive odd integer is of the form 6q+1,6q+3 or 6q+5