If n is an odd integer, then show that n2 – 1 is divisible by 8.

1.	If n is an odd integer, then show that n2 – 1 is divisible by 8.
Answer» We know that any odd positive integer n can be written in form 4q + 1 or 4q + 3. So, according to the question, 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.

Answer»

We know that any odd positive integer n can be written in form 4q + 1 or 4q + 3.

So, according to the question,

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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