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

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.
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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

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

I am not clear with the solution.

nk+4 = 24
nk = 20

if k=1, n=20
if k=2, n=10
if k=4, n=05

But why are we not considering the following cases too:
if k=5, n=04
if k=10, n=02
if k=20, n=01

Please clarify my doubt.

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

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

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?

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
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Is this really a level 700 question?

In the first post, you can find information about the difficulty level and user statistics of a question. For this particular question, its difficulty level is 700-Level. The system calculates the difficulty level based on the timer statistics of users who attempted the question.
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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?

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.
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