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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