n – 1 a multiple of 3?

[Needgmat]

Is positive integer n - 1 a multiple of 3?

(1) n^3 - n is a multiple of 3

(2) n^3 + 2n^2+ n is a multiple of 3

OAB

Please explain.

Many thanks in advance.

Kavin

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Hi Kavin,

This question can be solved in a couple of different ways, but you might find it easiest to combine some Algebra skills with TESTing VALUES (and some Number Property rules).

We're told that (N - 1) is a POSITIVE INTEGER, which means that N >= 2. We're asked if (N - 1) is a multiple of 3. This is a YES/NO question.

1) (N^3 - N) is a multiple of 3

IF....
N = 2
(N^3 - N) = 8 - 2 = 6
(N - 1) = 1 and the answer to the question is NO.

IF....
N = 4
(N^3 - N) = 64 - 4 = 60
(N - 1) = 3 and the answer to the question is YES.
Fact 1 is INSUFFICIENT.

2) (N^3 + 2N^2 + N) is a multiple of 3

It's interesting that the three 'terms' all contain an "N"; that hints that maybe we can factor this down...

N^3 + 2N^2 + N =
N(N^2 + 2N + 1) =
N(N+1)(N+1)

So now we know that N(N+1)(N+1) is a multiple of 3. For that product to be a multiple of 3, either (N) or (N+1) MUST be a multiple of 3.

If (N) is a multiple of 3 (re: 3, 6, 9, 12, etc.), then (N-1) would be 1 less (re: 2, 5, 8, 11, etc.), so it would NEVER be a multiple of 3 and the answer to the question would be NO.

If (N+1) is a multiple of 3 (re: 3, 6, 9, 12, etc.), then (N-1) would be 2 less (re: 1, 4, 7, 10, etc.), so it would NEVER be a multiple of 3 and the answer to the question would be NO.
Fact 2 is SUFFICIENT

Final Answer: B

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

[Matt@VeritasPrep]

This is much more legible using superscripts.

S1:

n³ - n =

n * (n² - 1) =

n * (n + 1) * (n - 1)

Since this is the product of three consecutive integers, it MUST be divisible by 3 in ANY CASE. This means that telling us "n³ - n is divisible by 3" is useless: we know that already, without the statement!

Since S1 is useless, the answer is B or E, so we can consider S2 solo.

S2:

n³ + 2n² + n is a multiple of 3

n * (n² + 2n + 1) is a multiple of 3

n * (n + 1)² is a multiple of 3

So either n is a multiple of 3 or (n + 1) is a multiple of 3. This means that (n - 1) CANNOT be a multiple of 3, since it's either 1 or 2 less than a multiple of 3, and our answer is B.