Question

using Euclid division lemma prove that if n is an odd positive integer then n^2-1 is divisible by 8​

Answer 1

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Answer 2

Given : n is an odd positive integer

To prove : n²-1 is divisible by 8

Explanation:

we know that for any odd integer is in the form of 4q+1 or 4q+3  for some integer q

case 1: when n = 4q+1

in this case we have

n² -1 = (4q+1)² -1

= 16q²+8q+1-1

= 8q(2q+1)

n²-1 is divisible by 8 since 8q(2q+1)  is also divisible by 8

case 2:

when n = 4q+3

n² -1 = (4q+3)² -1

= 16q²+24q+9-1

= 8(2q² +3q+1)

n²-1 is divisible by 8 since 8(2q² +3q+1)  is also divisible by 8

hence ,

n²-1 is divisible by 8

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