If n is an integer greater than 7, which of the following must be divi

Question

If n is an integer greater than 7, which of the following must be divisible by 3?

(A) n(n-5)(n-2)

(B) n(n+5)(n+3)

(C) n(n+2)(n-1)

(D) n(n-4)(n-2)

(E) n(n-7)(n+3)

Abhishek009 · Accepted Answer

Rule of divisibility states that a number is divisible by 3 if the sum of the digits is divisible by 3 , check for the same in the answer choices -

(A) n(n-5)(n-2)

=> n + (n-5) + (n-2) = 3n - 7 { Not divisible by 3 }

(B) n(n+5)(n+3)

=> n + (n+5) + (n+3) = 3n + 8 { Not divisible by 3 }

(C) n(n+2)(n-1)

=> n + (n+2) + (n-1) = 3n + 1 { Not divisible by 3 }

(D) n(n-4)(n-2)
 
 => n + (n-4) + (n-2) = 3n - 6 {Divisible by 3 }
 
 (E) n(n-7)(n+3)
 
 => n + (n-7) + (n+3) = 3n - 4 { Not divisible by 3 }

Hence correct answer will be (D)

FacelessMan · Answer

To solve a problem like this, we need to make sure that the number is a multiple of 3. And for us confirm that, we will have to check the remainder when these numbers are divided by 3. If in this process we can confirm that 3 numbers leave 3 different remainder, then one of the numbers will be a multiple of 3.

Let us start with C -> n(n+2)(n-1) = n(n+2)(n-3+2). Thus we can see that n+2 and (n-3+2) will leave the same remainder. But we need 3 terms with 3 different remainders.

Now lets look at D -> n(n-4)(n-2) = n(n-3 -1)(n-2). This represents 3 terms with 3 different remainders, and one of them will be divisible by 3.

Hence D.

Divyadisha · Answer

Multiplication of three consecutive integers are always multiple of 3

So either n (n+1) (n+2) or n(n-1) (n-2) is multiple of 3

A and D have (n-2) in option, so lets check for these two first.

we can add -3 to (n-1) to make it 3rd consecutive number i.e n-4

so n(n-4)(n-2) or option D should be the answer

AbdurRakib · Answer

My Approach : Number picking (I came up with selecting correct answer using this approach)

According to the question Integer greater than 7 could be n=8,9,10,11 etc.I chose from 8 to plug in and found when n=10 then Answer Choice A is not divisible by 3.So,on B,C,D,E plunging in number n=8,9 and 10 on each Answer choices only Answer Choice D is divisible by 3 for n=8,9 and 10. So the Correct answer is D.

For further Clarity,For bigger number of n(n=110,111,112,etc) If you use the Divisibility Rule,you'll see Answer Choice D is divisible by 3.
 Divisibility Rule: If the sum of the digits of each number is divisible by 3, the number is also divisible by 3

vitaliyGMAT · Answer

Hi Abhishek009

We can't use that approach in such type of questions.

First of all, adding those multiples won't give you sum of digits of the final product.

Consider the following case:

n(n + 6)(n - 3)

If we add we'll get: n + n + 6 + n - 3 = 3n + 3 = 3(n + 1). At first glance seems to be multiple of 3 but let's pick some number, say n=5

5 + 5 + 6 + 5 - 3 = 18 clearly divisible by 3, but actual product: 5*11*2 is not.

The problem is when we divide our product n(n + 6)(n - 3) by 3, our remainder will be defined by n^3/3 (6 and 3 will give us 0 as a remainder). So the question boils down to the fact: if n itself is a multiple of 3 or not, n+6 and n-3 won't give us any additional information to answer it.

In that particular question it was just a coincidence that we get right answer, in general - we won't. I think checking remainders is the most appropriate way to tackle such type of problems.

Hope that's clear.

Regards
Vitaliy

Heseraj · Answer

Well, i chose two numbers to be on the safe side. I chose number 8 and number 13, one even number and one odd but also a prime number. I have a feeling that when questions with such arrangements comes, normally prime numbers have hidden roles to play.

Mine came out to be D as well.

Mine came out to be D as well.

Babineaux · Answer

Every three consecutive integers is divisible by 3,

Going through the options, D is only three consecutive integers separated by -2.

Thus D, n(n-2)(n-4) is divisible by 3.

IMO ans is D.

fmcgee1 · Answer

This doesn't seem right at all.

Of course 5(5+6)(5-3) is not divisible by 3. It's not supposed to be. But the actual number itself, which is 5112, IS divisible by 3.

The sum of the digits of (5)(11)(2) is divisible by 3, and the number that it actually represents (5112 NOT 5*11*2*) is also divisible by 3).

messi16 · Answer

To do this question, simply add up and see if divisible by 3, as we know that if the sum of its digits is divisible by 3, then the number is divisible by 3. Thus, answer choice D, when added up, gives us 3n-6, which is divisible by 3 and thus is our answer choice.

GMATinsight · Answer

Let's Just substitute x = {8, 10, 11..}

@x = 8 
(A) n(n-5)(n-2) = 8*3*6 YES

(B) n(n+5)(n+3) = 8*13*11 NO

(C) n(n+2)(n-1) = 8*10*7 NO

(D) n(n-4)(n-2) = 8*4*6 YES

(E) n(n-7)(n+3) = 8*1*11 NO

So only options left are option A and D

@x = 10 
(A) n(n-5)(n-2) = 10*5*8 NO
(D) n(n-4)(n-2) = 10*6*8 YES

Hence Answer: Option D

If n is an integer greater than 7, which of the following must be divi

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If n is an integer greater than 7, which of the following must be divi

Prep Toolkit

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