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