Question

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

Answer 1

a positive integer is odd if it is of the form

"2p+1" where $P>0$

6q + 1 = 2(3q) + 1 $\Rightarrow$P = 3q

6q+3 = 2 [3q+1]+1 $\Rightarrow$ P = 3q+1

6q+5 = 2 [3q+2]+1 $\Rightarrow$ P = 3q+2

$\therefore$ any positive add integer on be written of the from 6q+1, 6q+3 or 6q+5