Orange08 wrote:

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 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 3Since we cannot answer the

target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n³ + 2n² + n is a multiple of 3More 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 3Case 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 3IMPORTANT: 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 3So, it MUST be the case that

n - 1 is NOT a multiple of 3Since we can answer the

target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,

Brent

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