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If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r?

1) n is not divisible by 2.
2) n is not divisible by 3.

OA C

We have to determine whether (n - 1)(n + 1) is divided by 24.

Let's take each statement one by one.

1) n is not divisible by 2.

=> n is an odd number. Thus, (n - 1) and (n + 1) are even. Note that at least one between (n - 1) and (n + 1) must be divisible by 4, thus, (n - 1)*(n + 1) must be divisible by 8. Example: (2, 4); (4, 6); (6, 8), etc.

However, since we do not have any information whether (n - 1) or (n + 1) is divisible by 3, we cannot conclude whether (n - 1)*(n + 1) is divided by 24 (= 8*3). Insufficient. For example, if n = 23, we have (n - 1)*(n + 1) = 22*24, we have (n - 1)*(n + 1) is divided by 24; the answer is yes. However, if n = 3, we have (n - 1)*(n + 1) = 2*4, we have (n - 1)*(n + 1) not divided by 24; the answer is no.

No unique answer. Insufficient.

2) n is not divisible by 3.

=> One between (n - 1) and (n + 1) must be divisible by 3. Pick any positive integer that is not divisible by 3 from 1, 2, 3, 4, 5, 6, 7, 8, ..., you will find that one of the integers, either on the left of it or on the right of it is divisible by 3.

However, we do not know whether (n - 1)*(n + 1) is divisible by 8. For example, if n = 2, we have (n - 1)*(n + 1) = 2, we have (n - 1)*(n + 1) is not divided by 24; the answer is no. However, if n = 25, we have (n - 1)*(n + 1) = 24*26, we have (n - 1)*(n + 1) divided by 24; the answer is yes.

No unique answer. Insufficient.

(1) and (2) together

From (1), we know that (n - 1)*(n + 1) is divisible by 8 and from (2), we know that (n - 1)*(n + 1) is divisible by 3, thus, (n - 1)*(n + 1) is divisible by 24. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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