Show that any positive odd integer is of the form $6q+1$, or $6q+3$, or $6q+5$, where $q$ is some integer.

Using Euclid division algorithm, we know that $a=bq+r,$ $0\le r\le b$ ----(1)

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

Then, by Euclid’s algorithm, $a=6q+r$ for some integer $q\ge 0$, and $r=0,1,2,3,4,5$ ,$or$ $0\le r<6$.

Therefore, $a=6qor6q+1or6q+2or6q+3or6q+4or6q+5$

$6q+0:6$ is divisible by $2$, so it is an even number.

$6q+1:6$ is divisible by $2$, but $1$ is not divisible by $2$ so it is an odd number.

$6q+2:6$ is divisible by $2$, and $2$ is divisible by $2$ so it is an even number.

$6q+3:6$ is divisible by $2$, but $3$ is not divisible by $2$ so it is an odd number.

$6q+4:6$ is divisible by $2$, and $4$ is divisible by $2$ so it is an even number.

$6q+5:6$ is divisible by $2$, but $5$ is not divisible by $2$ so it is an odd number.

And therefore, any odd integer can be expressed in the form $6q+1or6q+3or6q+5$