Question

show that any positve odd integer can be written in the form 6m + 1,6m + 3 or 6m + 5 where
m is a positive integer.

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Answer 1

Let n be any arbitrary positive odd integer.

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

So, by Euclid's division lemma, we have

n = 6m + r, where 0 ≤ r < 6.

As 0 ≤ r < 6 and r is an integer, r can take values 0, 1, 2, 3, 4, 5.

⇒ n = 6m or n = 6m + 1 or n = 6m + 2 or n = 6m + 3 or n = 6m + 4 or n = 6m + 5

But n ≠ 6m or n ≠ 6m + 2 or n ≠ 6m + 4 (∵ 6m, 6m + 2, 6m + 4 are multiples of 2, so an even interger whereas n is an odd integer)

⇒ n = 6m + 1 or n = 6m + 3 or n = 6m + 5

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