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Which is the divisor of n?

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by AndrewRivera7793 » Thu Dec 25, 2014 12:51 pm
When the positive integer n is divided by 45, the remainder is 18. Which of the following must be a divisor of n?

A) 11

B) 9

C) 7

D) 6

E) 4

Is there a formula of some sort to solve this problem? Any help is appreciated.

Thanks,

-Andy
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by GMATGuruNY » Thu Dec 25, 2014 1:04 pm
AndrewRivera7793 wrote:When the positive integer n is divided by 45, the remainder is 18. Which of the following must be a divisor of n?

A) 11

B) 9

C) 7

D) 6

E) 4

Is there a formula of some sort to solve this problem? Any help is appreciated.

Thanks,

-Andy
When the positive integer n is divided by 45, the remainder is 18.
In other words, n is multiple of 45 plus 18:
n = 45a + 18, where a is a nonnegative integer.

Simplifying n = 45a + 18, we get:
n = 45a + 18 = 9(5a + 2).
Thus, n must be a multiple of 9.

The correct answer is B.

We can also TEST options for n.

If a=0, then n = 45a + 18 = 45*0 + 18 = 18.
Since 18 is not a multiple of 11, 7, or 4, eliminate A, C and E.
If a=1, then n = 45a + 18 = 45*1 + 18 = 63.
Since 63 is a not multiple of 6, eliminate D.

The correct answer is B.
Last edited by GMATGuruNY on Thu Dec 25, 2014 1:08 pm, edited 1 time in total.
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by Brent@GMATPrepNow » Thu Dec 25, 2014 1:05 pm
AndrewRivera7793 wrote:When the positive integer n is divided by 45, the remainder is 18. Which of the following must be a divisor of n?

A) 11
B) 9
C) 7
D) 6
E) 4
When it comes to remainders, we have a nice rule that says:

If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Here, we are told that n divided by 45 leaves a remainder of 18, so the possible values of n are: 18, 63, 108,... etc.

IMPORTANT: the question asks, "Which of the following must be a divisor of n?
So, let's test the smallest possible value of n, which is 18, and check the answer choices.

18 is NOT divisible by 11, 7 or 4, so we can ELIMINATE A, C and E.
So, the correct answer is either B or D

Now test the next possible value of n, which is 63.
63 is NOT divisible by 6, so we can ELIMINATE D

So, by the process of elimination, the correct answer is B

Cheers,
Brent
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by AndrewRivera7793 » Thu Dec 25, 2014 1:12 pm
Wow, Thank you both for the excellent methods to solve this problem! I've "Thanked" you both. Merry Xmas :)
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by Jeff@TargetTestPrep » Thu May 17, 2018 5:11 pm
AndrewRivera7793 wrote:When the positive integer n is divided by 45, the remainder is 18. Which of the following must be a divisor of n?

A) 11

B) 9

C) 7

D) 6

E) 4
We can find all possible values of n by adding the remainder 18 to integer multiples of 45. Thus, we see that n can be values such as 18 or 63 or 108.

The only common factors of these three numbers are 1 and 9. Since 1 is not in the answer choices, the correct answer must be 9.

Answer: B

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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