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!
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