n â€“ 1 a multiple of 3

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j_shreyans: Legendary Member; Posts: 510; Joined: Thu Aug 07, 2014 2:24 am; Thanked: 3 times; Followed by:5 members

n â€“ 1 a multiple of 3

by j_shreyans » Tue Jun 16, 2015 8:46 am

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

Statement 1 : n^3 - n is a multiple of 3

n(n+1)(n-1)/3 question rephrased .

So n must be multiple of 3 , therefore n =(3,6,9,12...........)

so the answer to the target question is NO. So this statement is sufficient.

OA B

Please advise and correct me if I am wrong.

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Source: Beat The GMAT — Data Sufficiency | Tue Jun 16, 2015 8:46 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Tue Jun 16, 2015 9:13 am

Is positive integer n-1 a multiple of 3?

(1) nÂ³ - n is a multiple of 3
(2) nÂ³ + 2nÂ² + n is a multiple of 3

Target question: Is positive integer n-1 a multiple of 3?

Statement 1: nÂ³ - n is a multiple of 3
Factor: nÂ³ - n = n(nÂ² - 1) = n(n-1)(n+1) = (n-1)(n)(n+1)
Notice that n-1, n and n+1 are three CONSECUTIVE INTEGERS.
IMPORTANT: Statement 1 is simply telling us that the product of 3 consecutive integers is divisible by 3. This is not new information. The product of ANY 3 consecutive integers will always be divisible by 3. In fact, there's a rule that says, "The product of n consecutive integers is divisible by n, n-1, n-2, . . . 2, 1"
Since statement 1 is just some rule that already exists in mathematics, we already knew this information BEFORE we even examined statement 1. So, there's no way that statement 1 could possibly add any information to help us answer the target question.
As such, statement 1 is NOT SUFFICIENT

Statement 2: nÂ³ + 2nÂ²+ n is a multiple of 3
Factor: nÂ³ + 2nÂ² + n = n(nÂ² + 2n + 1) = n(n+1)(n+1)
This means that EITHER n is a multiple of 3 OR n+1 is a multiple of 3.
Let's examine each possible case:
case a: If n is a multiple of 3, then we can find other multiples of 3 by adding or subtracting multiples of 3 to n. So, for example, n+3 and n+6 will be also be multiples of 3. Likewise, n-3 and n-6 will be also be multiples of 3. Since n-1 is just 1 less than n, n-1 cannot be a multiple of 3 .
case b: If n+1 is a multiple of 3, then n-1 cannot be a multiple of 3 , Since n-1 is just 2 less than n+1.
Since both possible cases yielded the same answer to the target question, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com