When 24 is divided by the positive integer n, the remainder is 4. Whic

Question

When 24 is divided by the positive integer n, the remainder is 4. Which of the following statements about n must be true?

I. n is even 
II. n is a multiple of 5 
III. n is a factor of 20

A) III only 
B) I and II only 
C) I and III only 
D) II and III only 
E) I, II, and III

Bunuel · Accepted Answer

When 24 is divided by the positive integer n, the remainder is 4. Which of the following statements about n must be true? 
 
I. n is even 
II. n is a multiple of 5 
III. n is a factor of 20

A) III only 
B) I and II only 
C) I and III only 
D) II and III only 
E) I, II, and III

When 24 is divided by the positive integer n, the remainder is 4:

24 = qn + 4, where q is a quotient, n is a divisor and 4 is the remainder.  Notice that the remainder is always less than the divisor, so n > 4.

qn = 20.

q = 1 and n = 20.

q = 2 and n = 10.

q = 4 and n = 5.

Answer: D.

CounterSniper · Answer

When 24 is divided by the positive integer n, the remainder is 4

can be written as

24=nk+4 
nk=20

n can take the following values
5,10,20

therefore D

ScottTargetTestPrep · Answer

We are given that when 24 is divided by n, the remainder is 4. Thus, we see that n can be 5, 10, or 20.

Of the answer choices, we see that n must be a multiple of 5 or a factor of 20.

Answer: D

dave13 · Answer

Bunuel hello

one question, if q = 1 and n = 20 , then it means that n is even, why then first option (I. n is even ) is not correct

we divide by even n which is 20 and remainder is 4 , so my question why to exclude option one in which n is even

thank you and have a good day

Bunuel · Answer

The question asks which of the following MUST be true not COULD be true. n is not necessarily even, it could also be odd (n = 5).

nit99 · Answer

what about 15 - multiple of 5 which gives remainder = 9 . why no one is considering 15 integer.

CrackverbalGMAT · Answer

In this question, we are trying to find out which statement ‘Must be true’ or always true. If a statement has to always be true, it has to hold true whatever case you take. Even if it fails on one, it means that the statement is not always true.

This is the basis for our strategy for ‘Must be true’ questions. We always try to make a statement FALSE, by taking simple cases and eliminate options. Whatever option is left should be the answer. However, in this question, we can actually solve the entire question based on concepts and won’t have to take recourse to cases. Let’s see how we can do that.

When 24 is divided by the positive integer n, the remainder is 4. This means, we can write 24 in terms of n as follows:
24 = n*k + 4, where k is the quotient in the division process. Remember that when you have the remainder given, the quotient will always be an integral value. Therefore, k is an integer.

Re-arranging the terms of the equation above, we have n*k = 20. This means that n is a factor of the number 20. So, the possible values for n are 1, 2, 4, 5, 10 and 20. 
But, when 24 was divided by n, the remainder was 4. This means that the value of n should be more than 4 since the remainder cannot be more than the divisor ever. 
Therefore, n can only be 5 or 10 or 20. As you see, when you divide 24 by any of these three numbers, the remainder will be 4.

From this, we can say that statement I is not always true since 5 is not even. But, clearly, statement II is definitely true since all of 5, 10 and 20 are multiples of 5.

Coming to statement III, this is something we established during our analysis of the question stem, isn’t it. So, statement III has to be true.

The correct answer option is D.

Hope that helps!

ScottTargetTestPrep · Answer

24/15 = 1 remainder 9, not remainder 4. That is why we do not consider 15.

karantambat · Answer

if 24 is divided by 28, a positive integer, the remainder should be 4. There is no specification that n is greater or smaller than 24. So why haven't we considered this option?

MBAB123 · Answer

If 24 is divided by 28, the remainder is 28 and not 4 as 28 does not divide 24 at all. However, if 28 is divided by 24 then the remainder would be 4 as 24 divides 28 once and leaves 4 as the remainder. Hope this helps!

CrackverbalGMAT · Answer

In a fraction x/y, if x is less than y, the remainder is x.

If 24 is divided by 28, the remainder is 24.

Arun Kumar

jaydavido · Answer

can you pls explain; if N is 5 then III wouldn't be correct because 5 isn't a factor of 20. and the question says must be true? so wouldn't only II be correct?

Bunuel · Answer

An integer a is a factor of an integer b if a/b is an integer. Since 20/5 is an integer, it can be concluded that 5 is indeed a factor of 20 (20 = 5*4). Positive factors of 20 are 1, 2, 4, 5, 10, and 20.

MyNameisFritz · Answer

Hello!

A great question, got it wrong though. Could somebody please explain to me why does the remainder has to be always less than divisor n?

Bunuel · Answer

Remainders 
     Theory   
 REMAINDERS 
 Remainders: Tips and hints 
 Divisibility and Remainders on the GMAT 
 Remainders on the GMAT 
 A Remainders Shortcut for the GMAT 
 All About Negative Remainders on the GMAT 
 A Tricky Question on Negative Remainders 
 Compilation of tips and tricks to deal with remainders 
 Cyclicity and the remainders on the GMAT 
 GMAT Question Patterns: Remainders 
     Questions   
 Units digits, exponents, remainders problems 
 DS Questions 
 PS Questions

For other subjects:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

soniasw16 · Answer

When n = 15, remainder is 9. Why isnt that considered?

Krunaal · Answer

n cannot be 15, it is clearly mentioned in the question that  "When 24 is divided by the positive integer n, the remainder is 4."  =>  24 = n * (quotient) + 4

=> n * (quotient) = 20

When quotient = 1, n = 20

When quotient = 2, n = 10

When quotient = 3, n = 20/3 => not possible since we are told  n is a positive integer .

When quotient = 4, n = 5

So, n can only take values 5, 10, and 20 .

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