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 integers​

Answer 1

Let a and b be two integers then according to Euclid's division lemma there exists q and r satisfying

a = bq + r

taking b = 6

a = 6q + r , 0 < r < q

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

a = 6q = 2 (3q)

and ,

a = 6q + 1

a = 6q + 2

a = 2 (3q + 1)

a = 6q + 3

a = 6q + 4

= 2 (3q+ 2)

and , a = 6q + 5

Thus , any positive odd integer is of the form 6q+1,6q+3,6q+5