Question

use euclid division lemma to 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 a be any positive integer and b be 6
by euclids lemma ,
a=bq+r and here r can be 0 to 5 so a = 6q or 6q+1 or 6q +2 ........ 6q +5 but here 6q , 6q+2,6q+4 are even thus odd +ve integers are of form 6q+1 ,6q+3 ,6q +5.