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Question
If n is an odd integer, then show that n2 – 1 is divisible by 8.
Solution
We know that any odd positive integer n can be written in form 4q + 1 or 4q + 3.
When n = 4q + 1,
Then n2 – 1 = (4q + 1)2 – 1
= 16q2 + 8q + 1 – 1
= 8q(2q + 1) is divisible by 8.
When n = 4q + 3
Then n2 – 1 = (4q + 3)2 – 1
= 16q2 + 24q + 9 – 1
= 8(2q2 + 3q + 1) is divisible by 8.
So, from the above equations, it is clear that
If n is an odd positive integer
n2 – 1 is divisible by 8.
Hence Proved.
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