How Many Integers in a Group of 5 Consecutive Positive Integers are Divisible by 4? GMAT Data Sufficiency

Question: How many integers in a group of 5 consecutive positive integers are divisible by 4?

1. The median of these numbers is odd
2. The average (arithmetic mean) of these numbers is a prime number

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“How many integers in a group of 5 consecutive positive integers are divisible by 4?”– is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken f0rom the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation

Approach Solution 1:

1. The median of these numbers is odd. The median of the set with odd number of terms is just a middle term, thus our set of 5 consecutive numbers is: {Odd, Even, Odd, Even, Odd}. Out of 2 consecutive even integers only one is a multiple of 4.
   Hence, this statement is sufficient
2. The average (arithmetic mean) of these numbers is a prime number. In any evenly spaced set the arithmetic mean (average) is equal to the median, so we have that mean = median = prime. Since it’s not possible that median = 2 = even, (in this case not all 5 numbers will be positive), then median = odd prime, and we have the same case as above.
   Hence, this statement is sufficient

Correct Answer: D

Approach Solution 2:

The question asks how many numbers in the set are divisible by 4. So, 2 is not divisible by 4. Hence in the set (1,2,3,4,5), we have only 1 number that is divisible by 4. Hence the statement is sufficient.

Both the statements 1 and 2 lead us to the conclusion that the consecutive set starts with an ODD integer. So, the set will be of the form (odd, even, odd, even, odd).

Hence there will be two even integers in the set and both will be the consecutive even integers. Out of any two consecutive even integers, only 1 is divisible by 4 (example: 2,4,6,8,10, etc.). So, we can conclude that there is only 1 number that is divisible by 4. Hence both the statements are sufficient.

Correct Answer: D

Approach Solution 3:

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 1, 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 onle 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 (1) is sufficient to say that the set of integers consists of ONE number that is divisible by 4. Possible answer options are A and D. Answer options B, C, and E can be eliminated.

From Statement 2, 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 s (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 number. This is equivalent to the data given in the first statement. Since statement 1 alone was sufficient, statement 2 alone will be sufficient. This is a very simple and logical conclusion.

Answer option A can be eliminated.

Correct Answer: D

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