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.