9. Prove that the square of any positive integer is of the form 5q or 5q + 1, 5q + 4 for some integer q.

Solution:

Let ‘a’ be any positive integer.

Then,

According to Euclid’s division lemma,

a=bq+r

According to the question, when b = 5.

   

When r = 0, we get, a = 5k 

   

When r = 1, we get, a = 5k + 1

   

   

When r = 2, we get, a = 5k + 2

   

   

When r = 3, we get, a = 5k + 3

   

= 5q + 4, where q = 5k² + 6k + 1

When r = 4, we get, a = 5k + 4

a² = (5k + 4)² = 25k² + 16 + 40k = 5(5k² + 8k + 3) + 1

   

= 5q + 1, where q = 5k² + 8k + 3

Therefore, the square of any positive integer is of the form 5q or, 5q + 1 or 5q + 4 for some integer q.