Question

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

Answer 1

Let a be any positive integer and b= 6 .Then by Euclid's algorithm a=6q+r for some integer q>0 and where 0<r<6 , the possible remainders are,1,2,3,4,5. So , a can be 6q or 6q+1or 6q+2or 6q+3 or 6q+4 or 6q+5.
However, since a is odd positive integer, therefore we do not consider the cases . 6q, 6q+2 and 6q+4.
Hence , any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5..