Question

use division algorithm to show that any positive odd integer is in the form of 6q + 1 or 6q + 3 or 6q + 5 where is some integer

Answer 1

it's so easy
let b = 6
r = 0,1,2,3,4,5..
according to Euclid division lemma any positive integer can be written as a=bq+r where as 0< r < b..
positive odd integer is :- 6q+1, 6q+3, 6q+5 ....

Answer 2

We know,

"For every intergers a and b there always exists another two integers q and r , such that a = bq + r and 0 ≤ r <b."

Taking b = 6
a = 6q + 0 or
a = 6q + 1 or
a = 6q + 2 or
a = 6q + 3 or
a = 6q + 4 or
a = 6q + 5

Here,
6q + 1 , 6q + 3 and 6q + 5 are odd positive integers.