Question

show that for odd positive integer tobe a perfect square it should be of the form 8k+1

Answer 1

Given:

Odd positive integers.

To Prove:

Odd positive integer to be a perfect square it should be of the form 8k+1

PROOF:

1)Any odd positive integer which is of the form 

           4m + 1   or   4m + 3    where m is any integer.

2)Let,

• M = 4m + 1

Squaring on both sides, we get:

• M² = (4m + 1)²

• M² = 16m² + 8m + 1

• M² = 8 m (2m + 1) + 1

• M² = 8 k + 1      where k = m (2m + 1) 

3)Now let,

• N = 4m + 3

Squaring on both sides, we get:

• N² =(4m + 3)²

• N² = 16m² + 24m + 9

• N² = 16m² + 24m + 8 + 1

• N² = 8 (2m² + 3m + 1) + 1

• N² = 8 k + 1    where k = 2m² + 3m + 1

Every  odd positive integer tobe a perfect square it should be of the form 8k+1.

Hence proved