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

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Gmat_mission: Legendary Member; Posts: 1622; Joined: Thu Mar 01, 2018 7:22 amFollowed by: 2 members

by Gmat_mission » Thu May 21, 2020 12:53 am

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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. $n(n+5)(n−6)$

[spoiler]OA=A[/spoiler]

Source: Official Guide

Source: Beat The GMAT — Problem Solving | Thu May 21, 2020 12:53 am

anonylfc: Junior | Next Rank: 30 Posts; Posts: 11; Joined: Wed May 20, 2020 9:34 pm

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

by anonylfc » Thu May 21, 2020 9:48 am

The basic concept here is to make three numbers, which have different remainders when dividing by three, meaning that one of them should have the remainder of 1, another the reminder of 2 and the third one the remainder of 0, so that atleast one of them is divisible by 3.

In A option n-4 will have the same remainder as n-1, we already have n and n+1, so basically we have 3 consecutive numbers, so atleast one of them has to be divisible by 3.

Just to explain in detail, in option B-> n (n+2) (n-1),
n-1 and n+2 will have the same remainder when divisible by 3 ( For eg - 6 and 9, 5 and 8 -> all have the same remainder when divided by 3)

So hence the equation basically boils down to (n+2)(n) but what if (n+1) is divisible by 3 ? Hence not sufficient.

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

by Scott@TargetTestPrep » Mon Jun 01, 2020 7:12 am

Gmat_mission wrote: ↑
Thu May 21, 2020 12:53 am
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. $n(n+5)(n−6)$

[spoiler]OA=A[/spoiler]

Source: Official Guide

Solution:

Since all the expressions have the factor n, if n is a multiple of 3, then any of the expressions is divisible by 3. So let’s assume that n is not a multiple of 3. That is, n is either 3k + 1 or 3k + 2 for some integer k.

If n = 3k + 1, then n - 4 = 3k - 3 = 3(k - 1), n - 1 = 3k, and n + 5 = 3k + 6 = 3(k + 2) are all multiples of 3. This eliminates choice C and D.

If n = 3k + 2, then only n + 1 = 3k + 3 = 3(k + 1) is a multiple of 3. Therefore, choice A is the correct answer since no matter what n is, one of its factors will be a multiple of 3.

Answer: A

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