Question

show that any positive odd intiger is of the form 6q+1,6q+3 or 6q+5, where q is some integer

Answer 1

Let a be any positive integer and b=6
By using the euclids lemma we get
a=bq+r.
a=6q+r

If we take r=0 ,a= 6q even
r=1 , a=6q+1 ( odd)
r=2 ,a =6q+2 (even)
r=3 ,a=6q+3 (odd)
r=4 ,a=6q+4 (even)
r=5 ,a=6q+5 (odd)

Therefore 6q+1,6q+3 and 6q+5 are odd
Therefore for any positive odd integer we get write 6q+1, 6q+3,and6q+5