Question

show that any positive odd integer is of the form 6q + 1, 6q + 5 where q is some integer

Answer 1

This is the solution

Answer 2

Let a be any positive odd integer.

a=bq+r
a=6q+r as b=6

0≤r<6
The possible values of r are 0,1,2,3,4 and 5

a=1,3,5,7,9....

1=6(0)+1 (r=1)
3=6(0)+3 (r=3)
5=6(0)+5 (r=5)
7=6(1)+1 (r=1)
9=6(1)+3 (r=3)
11=6(1)+5 (r=5)

Therefore,any positive odd integer is of the form 6q+1,6q+3 or 6q+5.

Hence proved.