Prove that if a positive integer is of the form 6q + 5, then it is of the form 3q + 2 for some integer q, but not conversely.

Question

Philoid · Accepted Answer

let A = 6q + 5, where q is a positive integer. We know that any positive integer is of the form 3m or, 3m + 1 or, 3m + 2.

Case 1:

A = 6q + 5 ⇒ 3× 2q + 3 + 2

⇒ 3(2q + 1) + 2 ⇒ 3N + 1; where N = 2q + 1 which is a positive integer

Case 2:

When A = 3q + 2 ⇒ 6 +5

Since is not always an integer.