Question

use euclid division lemma to show that any positive odd integer is form 6q+1, or 6q+3, or 6q+5,where q is integer​

Answer 1

let a be any positive integer and b =6

by using euclid's division lemma

a=bq+r

where

$0 \leqslant r < 6$

here r can be written as

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

so the required positive integer are 6q,6q+1,6q+2,6q+3,6q+4,6q+5

but

6q,6q+2,6q+4 are divisible by 2

so these are positive even integer

therefore

6q+1,6q+3,6q+5 are the positive odd integer