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.

Given: Integers of form 6q + 5 and 3q + 2.

To prove: positive integer is of the form 6q + 5, then it is of the form 3q + 2 for some integer q, but not conversely.

Explanation:

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

Since is not always an integer.