Question

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

Answer 1

Solution:
Any odd integer n is of the form 4m + 1 or 4m + 3.

=> n² – 1 = (4m + 1)² – 1
= 16m² + 8m
= 8(2m² + m),
which is divisible by 8.
Also, n² – 1 = (4m + 3)² – 1
= 16m² + 24m + 8
= 8(2m² + 3m + 1),
which is divisible by 8.
Hence, n² – 1 is divisible by 8 for any odd integer n.

Comments

Y are you taking some odd integers as 4m+1 or 4m+3.
Why no we take 2m+1 or 3m+1

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Namasthe,
Y are you taking some odd integers as 4m+1 or 4m+3.
Why no we take 2m+1 or 3m+1. Is there any specific reason for taking as 4m+1 or 4m+3.

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Because every odd integer will cover by 4m+1 or 4m+3.
You may also use 2m+1.