If a is an odd integer prove that a^2 − 1 is always divisible by 8.

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If a is divided by 4 then the possible set of remainders are 0, 1, 2 and 3.

Since a is odd so the remainder when divided by 4 cannot be 0 or 2.

So if a is an odd integer then the remainder when divided by 4 is 1 or 3.

Case 1: Let the remainder be 1.

So a = 4k + 1 for some integer k.

Thus a² = (4k + 1)² = 16k² + 8k + 1 = 8(2k² + k) + 1 So a² − 1 = 8(2k² + k) and so a² − 1 is divisible by 8

Case 2: Let the remainder be 3.

So a = 4k + 3 for some integer k.

Thus a² = (4k + 3)² = 16k² + 24k + 9 = 8(2k² + 3k + 1) + 1 So a² − 1 = 8(2k² + 3k + 1) and so a² − 1 is divisible by 8