Question

use euclid's division Lemma to show that the square of any positive integer is the form 3p , 3p +1

Answer 1

see the attachment for your answer

Answer 2

HI

Let a be any positive integer.

Therefore, a can be 3q or 3q+1 or 3q+2

where b = 3 and r = 0,1,2

(3q)² = 9q²

= 3(3q)²

= 3m ➡️ p = 3q²

(3q+1)² = (3q)² + 2(3q)(1) + (1)²

= 9q² + 6q + 1

= 3(3q² + 2q) + 1

= 3m + 1 ➡️p = 3q² + 2q

(3q+2)² = (3q)² + 2(3q)(2) + (2)²

= 9q² + 12q + 4

= 9q² + 12q + 3 + 1

= 3(3q² + 4q + 1) + 1

= 3m + 1 ➡️p = 3q² + 4q + 1

Therefore, the square of any positive integer is of the form 3p or 3p+1.

Hope it proved to be beneficial....