Question

using euclid's division Lemma Prove the difference of squares of two odd natural numbers is multiple of 8​

Answer 1

Given :  difference of squares of two odd natural numbers

To find : Prove its divisible by 8

Solution:

Let say two odd numbers are

2a + 1    & 2b + 1    a  > b

(2a + 1)² - (2b + 1)²

using x² - y²  = (x + y)(x - y)

= (2a + 1 + 2b + 1)(2a + 1 - 2b - 1)

= (2a + 2b + 2)(2a - 2b)

= 2(a + b + 1)2(a - b)

= 4(a + b + 1)(a - b)

if a & b both are even   then a - b is even

hence 4 * 2   = 8

Divisible by 8

if a & b both are odd   then a - b is even

hence 4 * 2   = 8

Divisible by 8

if in a & b one is odd and another is even

then a + b + 1  is even

hence 4 * 2   = 8

Divisible by 8

Hence proved that difference of squares of two odd natural numbers is multiple of 8​

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