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If n is an integer greater than 6, which of the following

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If n is an integer greater than 6, which of the following

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If n is an integer greater than 6, which of the following must be divisible by 3 ?

(A) n(n + 1)(n - 4)
(B) n(n + 2)(n - 1)
(C) n(n + 3)(n - 5)
(D) n(n + 4)(n - 2)
(E) n(n + 5)(n - 6)

OA A

Source: Official Guide

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BTGmoderatorDC wrote:
If n is an integer greater than 6, which of the following must be divisible by 3 ?

(A) n(n + 1)(n - 4)
(B) n(n + 2)(n - 1)
(C) n(n + 3)(n - 5)
(D) n(n + 4)(n - 2)
(E) n(n + 5)(n - 6)

OA A

Source: Official Guide
We can let n = 7 and then analyze each answer choice:

A) 7(8)(3) is divisible by 3.

B) 7(9)(6) is divisible by 3.

C) 7(10)(2) is not divisible by 3.

D) 7(11)(5) is not divisible by 3.

E) 7(12)(1) is divisible by 3.

We can eliminate choices C and D.

Now let n = 8 and analyze the remaining 3 choices:

A) 8(9)(4) is divisible by 3.

B) 8(10)(7) is not divisible by 3.

E) 8(13)(2) is not divisible by 3.

We see that the correct choice must be A.

Alternate Solution:

Let’s analyze each answer choice:

A) If n is divisible by 3, then the product n(n + 1)(n - 4) is also divisible by 3. If n produces a remainder of 1 when divided by 3 (for example, n = 7), then n - 4 must be divisible by 3. Finally, if n produces a remainder of 2 when divided by 3, n + 1 is divisible by 3. In either case, we see that the product has a factor that is a multiple of 3; therefore the product is always divisible by 3.

Answer: A

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Among any three consecutive integers, you always have exactly one multiple of 3. So if one of our answers here was something like (n-1)(n)(n+1), the product of three consecutive integers, it would certainly be divisible by 3. We don't have something quite that easy, but if you look at answer A:

(n-4)(n)(n+1)

notice that n-4 and n-1 are exactly three apart, so if n-4 is divisible by 3, so is n-1. So (n-4)(n)(n+1) is divisible by 3 exactly when (n-1)(n)(n+1) is, and since (n-1)(n)(n+1) always is, A is the right answer.

But I think most people solve this question by testing numbers, which is a viable approach here.

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BTGmoderatorDC wrote:
If n is an integer greater than 6, which of the following must be divisible by 3 ?

(A) n(n + 1)(n - 4)
(B) n(n + 2)(n - 1)
(C) n(n + 3)(n - 5)
(D) n(n + 4)(n - 2)
(E) n(n + 5)(n - 6)
One approach is to test values

If n = 7, we get:
(A) 7(7 + 1)(7 - 4) = (7)(8)(3). Definitely divisible by 3. KEEP
(B) 7(7 + 2)(7 - 1) = (7)(9)(6). Definitely divisible by 3. KEEP
(C) 7(7 + 3)(7 - 5) = (7)(10)(4). NOT divisible by 3. ELIMINATE.
(D) 7(7 + 4)(7 - 2) = (7)(11)(5). NOT divisible by 3. ELIMINATE.
(E) 7(7 + 5)(7 - 6) = (7)(12)(1). Definitely divisible by 3. KEEP

We're left with A, B, and E

Try n = 8. We get:
(A) 8(8 + 1)(8 - 4) = (8)(9)(4). Definitely divisible by 3. KEEP
(B) 8(8 + 2)(8 - 1) = (8)(10)(7). NOT divisible by 3. ELIMINATE.
(E) 8(8 + 5)(8 - 6) = (8)(13)(2). NOT divisible by 3. ELIMINATE.


Answer: A

Cheers,
Brent

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