Is positive integer n-1 a multiple of 3?

(1) n³ - n is a multiple of 3
(2) n³ + 2n² + n is a multiple of 3

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

Statement 1: n³ - n is a multiple of 3
Factor: n³ - n = n(n² - 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: n³ + 2n²+ n is a multiple of 3
Factor: n³ + 2n² + n = n(n² + 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

jaiyeolab wrote:I know how to solve the below using math. I was however curious to know whether it is possible to solve this question or something similar by testing values? I am training myself to try to solve most questions that include variables, by testing values.

Is positive integer n - 1 a multiple of 3?

(1) n^2 - n is a multiple of 3

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

Testing values is tricky for this question, plus you have the potential to choose bad numbers and draw an incorrect conclusion. See my article for more on this: https://www.gmatprepnow.com/articles/dat ... lug-values

Cheers,
Brent

Here's my general rule: if you can see and understand the underlying concept, that will likely be more effective that testing numbers. You should use number testing if you can't identify any particular rule or concept at play.

In this case, for example, recognizing that the question is testing products of consecutive integers would be much faster and easier than testing numbers. You can quickly ID everything you know about the product of 3 consecutives: it will always be even and it will always be a multiple of 3, etc.

Testing numbers is most effective on topics like digit placement (if the digits in a certain 2-digit number were reversed...) in which there is no universal rule to apply. On topics such as divisibility, odds/evens, or positives/negatives, it's best to learn the underlying concepts well, then commit the rules to memory.