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

Is positive integer n – 1 a multiple of 3?

(1) n^3 – n is a multiple of 3 --> \(n^3-n=n(n^2-1)=(n-1)n(n+1)=3q\). Now, \(n-1\), \(n\), and \(n+1\) are 3 consecutive integers and one of them must be multiple of 3, so no wonder that their product is a multiple of 3. However we don't know which one is a multiple of 3. Not sufficient.

(2) \(n^3 + 2n^2+ n\) is a multiple of 3 --> \(n^3 + 2n^2+ n=n(n^2+2n+1)=n(n+1)^2=3p\) --> so either \(n\) or \(n+1\) is a multiple of 3, as out of 3 consecutive integers \(n-1\), \(n\), and \(n+1\) only one is a multiple of 3 then knowing that it's either \(n\) or \(n+1\) tells us that \(n-1\) IS NOT multiple of 3. Sufficient.

Answer: B.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Out of any 3 consecutive integers only 1 is a multiple of 3. We know it's either n or n + 1, therefore it's not n-1.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Hello,

What is the approximate level of this question on 800 scale?

Regards.

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

Statement 1: n³ – n is a multiple of 3
Let's do some FACTORING
n³ – n = n(n² - 1) = n(n + 1)(n - 1)
So, statement 1 tells us that n(n + 1)(n - 1) is a multiple of 3
Notice that (n-1), n and (n+1) are 3 CONSECUTIVE integers, and we know that 1 out of every 3 consecutive integers is a multiple of 3 (e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, etc.
So, ONE of the following n, (n + 1), or (n - 1) a multiple of 3, but WHICH ONE??
It COULD be the case that (n - 1) IS a multiple of 3, OR it could be the case that (n - 1) is NOT a multiple of 3
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n³ + 2n² + n is a multiple of 3
More FACTORING
n³ + 2n² + n = n(n² + 2n + 1)
= n(n + 1)(n + 1)
So, statement 2 tells us that n(n + 1)(n + 1) is a multiple of 3
This means that EITHER n is a multiple of 3 OR (n + 1) is a multiple of 3
Let's examine both cases:

Case a: n is a multiple of 3
If n is a multiple of 3, then n - 1 CANNOT be a multiple of 3.
Why not?
Well, we already know that 1 out of every 3 consecutive integers is a multiple of 3
So, if n is a multiple of 3, the n+3 is also a multiple of 3 AND n+6 is a multiple of 3, AND n+9 is a multiple of 3, etc
Likewise, n-3 is a multiple of 3 AND n-6 is a multiple of 3, AND n-9 is a multiple of 3, etc.
Notice that n-1 is NOT among the possible multiples of 3
So, in this case, the answer to the target question is NO, n - 1 is NOT a multiple of 3

Case b: (n + 1) is a multiple of 3
If n+1 is a multiple of 3, then n - 1 CANNOT be a multiple of 3.
We'll use the same logic we used in case a above
1 out of every 3 consecutive integers is a multiple of 3
So, if n+1 is a multiple of 3, the n+1+3 (aka n+4) is also a multiple of 3 AND n+1+6 (aka n+7) is a multiple of 3, AND n+1+9 (aka n+10)is a multiple of 3, etc
Likewise, n+1-3 (aka n-2) is a multiple of 3 AND n+1-6 (aka n-5) is a multiple of 3, AND n+1-6 (aka n-5) is a multiple of 3, etc.
Notice that n-1 is NOT among the possible multiples of 3
So, in this case, the answer to the target question is NO, n - 1 is NOT a multiple of 3

IMPORTANT: for statement 2, there are only two possible cases, and in each case, the answer to the target question is the SAME: NO, n - 1 is NOT a multiple of 3
So, it MUST be the case that n - 1 is NOT a multiple of 3
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent
_________________

Brent Hanneson – Founder of gmatprepnow.com