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?Number plugging method:

(n-1)(n+1)=n^2-1(1) n is not divisible by 2 --> pick two odd numbers: let's say 1 and 3 --> if

n=1, then

n^2-1=0 and as zero is divisible by 24 (zero is divisible by any integer except zero itself) so remainder is 0 but if

n=3, then

n^2-1=8 and 8 divided by 24 yields remainder of 8. Two different answers, hence not sufficient.

(2) n is not divisible by 3 --> pick two numbers which are not divisible by 3: let's say 1 and 2 --> if

n=1, then

n^2-1=0, so remainder is 0 but if

n=2, then

n^2-1=3 and 3 divided by 24 yields remainder of 3. Two different answers, hence not sufficient.

(1)+(2) Let's check for several numbers which are not divisible by 2 or 3:

n=1 -->

n^2-1=0 --> remainder 0;

n=5 -->

n^2-1=24 --> remainder 0;

n=7 -->

n^2-1=48 --> remainder 0;

n=11 -->

n^2-1=120 --> remainder 0.

Well it seems that all appropriate numbers will give remainder of 0. Sufficient.

Algebraic approach:

(1) n is not divisible by 2. Insufficient on its own, but this statement says that

n=odd -->

n-1 and

n+1 are consecutive even integers -->

(n-1)(n+1) must be divisible by 8 (as both multiples are even and one of them will be divisible by 4. From consecutive even integers one is divisible by 4: (2, 4); (4, 6); (6, 8); (8, 10); (10, 12), ...).

(2) n is not divisible by 3. Insufficient on its own, but form this statement either

n-1 or

n+1 must be divisible by 3 (as

n-1,

n, and

n+1 are consecutive integers, so one of them must be divisible by 3, we are told that it's not

n, hence either

n-1 or

n+1).

(1)+(2) From (1)

(n-1)(n+1) is divisible by 8, from (2) it's also divisible by 3, therefore it must be divisible by

8*3=24, which means that remainder upon division

(n-1)(n+1) by 24 will be 0. Sufficient.

Answer: C.

Hope it helps.

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