M12-21

This is an excellent question on statistics and number properties. From the question statement, we know that the integers are consecutive and that they are all positive integers.

In a group of n consecutive integers, one integer will always be a multiple of n. For example, in a set of 3 consecutive integers, there will always be a multiple of 3. This is the underlying concept that we can use to solve this question.

However, we need to be careful about the possibility that there could be two multiples of 4 in a set of 5 consecutive integers. For example, 4,5,6,7,8 contains two integers which are divisible by 4. This is where we use the statements to evaluate how many of them can be divisible by 4.

Let us represent the 5 consecutive integers as a, (a+1), (a+2), (a+3) and (a+4). Remember that we have already taken these in order since these are consecutive.

From statement I, the median of these numbers is odd. This means that (a+2) is odd, which means that there are 3 odd numbers and 2 even numbers. In this case, only one of the even numbers will be divisible by 4. A set of 5 numbers will have 2 multiples of 4 when the first number itself is a multiple of 4, because of the rule stated above.. Clearly, that’s not happening here.

Statement I is sufficient to say that the set of integers consists of ONE number that is divisible by 4. Possible answer options are A or D. Answer options B, C and E can be eliminated.

From statement II, the average of the given numbers is a prime number.


For equally spaced values in a data set, Mean = Median.
Consecutive integers definitely represent equally spaced values. So, for the numbers that we have considered, the average(mean) is going to be the middle value.

So, the average is (a+2).
If(a+2) = 2, a = 0 which is not possible because all the numbers in the set are positive integers.

Therefore, (a+2) has to be an odd prime number. This is equivalent to the data given in the first statement. Since statement I alone was sufficient, statement II alone will also be sufficient. This is a very simple and logical conclusion.

Answer option A can be eliminated. The correct answer option is D.

Hope that helps!

Hi,
In all cases, any number, n, will have only one number divisible by itself in 'n' consecutive numbers..
5 will have only one multiple in any 5 consecutive number because of the property of multiplication tables..

I understood solution, but I just wanted to know why exactly does 1 of the number will be divisible by 4? Is their any kind of rule here that i have missed. I took various different sets of positive consecutive numbers with median being odd and in all cases,one of the number was divisible by 4.

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

actually, I am not able to see the options related to the question, I can just see is the question and 2 statement. how you are getting to the option D please help .

Hi,

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

I suggest you to go through the following post ALL YOU NEED FOR QUANT.

Hope this helps.

Let's analyze each statement separately:
To summarize: (1) The median of the set is odd: True. The pattern {Odd, Even, Odd, Even, Odd} guarantees that there is exactly one integer divisible by 4, making the median odd. (2) The average of the set is a prime number: True. The mean and median are the same in an evenly-spaced set, and since the median is an odd prime, the mean is also an odd prime.
Both statements hold true for any set of 5 consecutive positive integers. The answer is (D)

The assumption that the set only contains O,E,O,E,O is erroneous because another set can be E,O,E,O,E and there also the set starting with the multiple of 4 only has two integers divisible by 4

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You are mistaken.

For (1), the set {Even, Odd, Even, Odd, Even} is not valid because the median, which is the middle number, would be even, not odd as given in (1).

For (2), the set {Even, Odd, Even, Odd, Even} is not possible because it would require the average, which is the middle number, to be an even prime. There is only one even prime number, which is 2. However, the set cannot be {Even, Odd, 2, Odd, Even} because that would result in {0, 1, 2, 3, 4}, which is not valid since all the integers in the set must be positive, and 0 is not positive.

I would advise you to carefully re-read the question and the solution.

Isn't the question written incorrectly? What am I missing? Shouldn't it state that we are seeking where there is one integer divisible by 4?

How many integers in a set of 5 consecutive positive integers are divisible by 4?

My initial interpretation (without reviewing the two proof statements) is that it can be either one or two numbers divisible by 4 in a set of 5 consecutive integers.

Cases:
1.) {2,3,4,5,6} 4 is divisible by 4
2.) {16,17,18,19,20} 16 and 20 are divisible by 4

Then we have the statements
(1) The median of the set is odd.
Yes enough evidence for 1 integer divisible by 4, but not 2.
(2) The average (arithmetic mean) of the set is a prime number.
Yes enough evidence for 1 integer divisible by 4, but not 2.

The question asks: How many integers in a set of 5 consecutive positive integers are divisible by 4?

In a set of 5 consecutive positive integers there can be 1 or 2 integers divisible by 4. So, the question essentially asks which of these two cases we have.

(1) and (2) individually confirm that we have the first case. So, not really sure what is your question there?