Question

Show that any positive odd integer is of the form (6m + 1) or (6m + 3) or (6m + 5), where m is some integer.

Answer 1

Let s be any positive odd integer.

On dividing s by 6, let m be the quotient and r be the remainder.

By Euclid’s division lemma,

s = 6m + r, where 0 ≤ r ˂ 6

So we have, s = 6m or s = 6m + 1 or s = 6m + 2 or s = 6m + 3 or s = 6m + 4 or s = 6m + 5.

6m, 6m + 2, 6m + 4 are multiples of 2, but s is an odd integer.

Again, s = 6m + 1 or s = 6m + 3 or s = 6m + 5 are odd values of s.

Thus, any positive odd integer is of the form (6m + 1) or (6m + 3) or (6m + 5), where m is any odd integer.