Question

use Euclid division lemma to show that any positive odd integer is of the form of 6q+1,6q+3,6q+5​

Answer 1

let n be a given positive odd integer.

On dividing n by 6, let q be the quotient and r be the remainder.

By Euclid's division lemma,

n = 6q+r , where 0≤ r ≤6

.°. n = 6q or (6q+1) or (6q+2) or (6q+3) or (6q+4) or (6q+5).

But, n= 6q , n= (6q+2), n = (6q+4) give even values for n.

Thus, when n is odd, it is of the form (6q+1) or (6q+3) or (6q+5)