Question

Show that any positive integers is of the form 3 m or (3m+1) or (3m+2) or some integer m.

Answer 1

Let a be a positive integer.

Then, Using Euclid's Division Lemma,

a = bq + r, 0 <= r <b, b = 3

Then, r = 0, 1, 2

If r = 0,

a = 3q+0 = 3q

If r = 1,

a = 3q + 1

If r = 2,

a = 3q + 2

In all the above cases, m = q.

Hence, any positive integer is of the form 3m, 3m + 1 or 3m + 2 for some integer m.

Answer 2

Let a be any positive integer and b = 3. Then by Euclid's division lemma, we know that

a = bq + r, where 0 ≤ r < 3

∴ r = 0, 1 or 2

When r = 0,

a = 3q + 0 = 3q, where m = q.

When r = 1,

a = 3q + 1, where m = q.

When r = 2,

a = 3q + 2, where m = q.

Therefore, any positive integer can be of the form 3m or (3m+1) or (3m+2) for some integer m.