Divisibility

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

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

Answer -C

[mevicks]

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

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

Answer -C

Given:
n > 0 and is a +ive integer (n = 1, 2, 3 ...)
(n-1)(n+1) --> (n^2-1) when divided by 24 yields a remainder r, thus n^2-1 may or may not be divisible by 24.

r = ?

St1: n is not divisible by 2 (in other words n is not even). Test some numbers:
n = 1, n^2 -1 / 24 --> remainder = 0
n = 3, n^2 -1 / 24 --> remainder = 8
As we can't answer the target question definitely , INSUFFICIENT

St2: n is not divisible by 3. Test some numbers:
n = 1, n^2 -1 / 24 --> remainder = 0
n = 2, n^2 -1 / 24 --> remainder = 3
As we can't answer the target question definitely , INSUFFICIENT

St1+St2: We now know : n is not even, not divisible by 3 --> thus not divisible by 6.
Again test some numbers:
n = 1, n^2 -1 / 24 --> remainder = 0
n = 5, n^2 -1 / 24 --> remainder = 0
n = 7, n^2 -1 / 24 --> remainder = 0

Since in all the cases we get the remainder as 0, [spoiler]Answer = C[/spoiler]

Regards,
Vivek

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Hi shibsriz,

This DS question is perfect for TESTing values:

We're asked for the remainder when (n-1)(n+1) is divided by 24.

Fact 1: n is NOT divisible by 2.

This means that n must be ODD, so we have to restrict our TESTS to odd numbers
If n = 1, then (0)(1) = 0; 0/24 = 0 r 0
If n = 3, then (2)(4) = 8; 8/24 = 0 r 8
Different answers means Fact 1 is INSUFFICIENT

Fact 2: n is NOT divisible by 3

This means that n is NOT a multiple of 3.
If n = 1, then (0)(1) = 0; 0/24 = 0 r 0
If n = 2, then (2)(3) = 6; 6/24 = 0 r 6
Different answers means Fact 2 is INSUFFICIENT

Together, n is NOT divisibly by 2 nor 3, which severely restricts the possibilities:
If n = 1, then (0)(1) = 0; 0/24 = 0 r 0
If n = 5, then (4)(6) = 24; 24/24 = 1 r 0
If n = 7, then (6)(8) = 48; 48/24 = 2 r 0
Answers are consistent, so combined it's SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich