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If n is a positive integer and r is the remainder when

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If n is a positive integer and r is the remainder when

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If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?

(1) n is odd
(2) n is not divisible by 8

OA A

Source: Manhattan Prep

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BTGmoderatorDC wrote:
If n is a positive integer and r is the remainder when n^2 - 1 is divided by 8, what is the value of r?

(1) n is odd
(2) n is not divisible by 8
What is the remainder when (n+1)(n-1) is divided by 8?

Statement 1:
Since n is odd, both n-1 and n+1 are EVEN, implying that (n-1)(n+1) = the product of two consecutive even integers.
Given any two consecutive even integers, one will be a multiple of 4, while the other will be an even non-multiple of 4.
Since (multiple of 4)(even) = (multiple of 8), the product of any two consecutive integers will always be a multiple of 8.
Thus, when (n-1)(n+1) is divided by 8, the remainder will be 0.
SUFFICIENT.

Statement 2:
If n=1, then (n²-1)/8 = 0/8 = 0 R0.
If n=2, then (n²-1)/8 = 3/8 = 0 R3.
Since the remainder can be different values, INSUFFICIENT.

The correct answer is A.

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