Question

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

Answer 1

Here, the integer ‘n’ is of the form 5q+1.

⇒ n= 5q+1 

On squaring it, 

⇒ n²= (5q+1)² 

⇒ n²= (25q²+10q+1) 

⇒ n²= 5(5q²+2q)+1 

⇒ n²= 5m+1, where m is some integer. [For m = 5q²+2q] 

Therefore, the square of any positive integer of the form 5q + 1 is of the same form.