Question

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

Answer 1

let n be a given possitive odd integer
on dividing n by 6 let m be the quotient and r be the reminder we have
n=6q+r where 0 <=r <6
n=6m+r where r=0 1 2 3 4 5
n=6q or 6q+1 or 6q+2 6q+3 6q+4 6q+5
but n= 6q .....2 4 give even value of n.
thus when n is odd it is of the form 6q+1,3,5 for some integer q.

Answer 2

Let a be any positive odd integer and b =6

By using euclids lemma we get
a=6q+r

When r=0,1,2,3,4,5,

We have, r=0, a=6q. ( Even)

When, 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
Is even.

So for any positive odd integer we can write 6q+1 , 6q+3 and 6q+5.