Q6 of 124 Page 2

Prove that the square of any positive integer of the form 5q + 1 is of the same form.

Let N = 5p + 1. Then,

According to the condition:

N² = 25p² + 10p + 1 ⇒ 5(5p² + 2p) + 1 ⇒ 5A+1

Where A = 5p² + 2p

Therefore N² is of the form 5m + 1.

For any positive integer, prove that divisible by 6.

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.

Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m + 2.

Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.