If N different positive integers are added and the sum is denoted as S, and if S=NX, is X an integer?
(1) N is odd.
(2) All N numbers are consecutive integers.
OA C
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If N different positive integers are added and the sum is
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In the notation used in the question, S/N = sum/number of terms = the average of the list. So the question is just asking "is the average of the N numbers an integer?"
Statement 1 tells us we have an odd number of integers, which is not sufficient, since we could have the list 1, 2, 3, with an average of 2, or the list 1, 2, 4, with an average of 7/3. So the average may or may not be an integer.
Statement 2 tells us our list consists of consecutive integers. Then our average might be an integer, if our list is 1, 2, 3, or might not be, if our list is 1, 2, so this is not sufficient.
Using both Statements, we have an odd number of consecutive integers. With an odd number of things, the median must be in our set, so the median of this set must be equal to one of the integers in the set, and the median is an integer. But in a set of consecutive integers, the average equals the median, so the average is also an integer, and the two Statements together are sufficient. C.
Statement 1 tells us we have an odd number of integers, which is not sufficient, since we could have the list 1, 2, 3, with an average of 2, or the list 1, 2, 4, with an average of 7/3. So the average may or may not be an integer.
Statement 2 tells us our list consists of consecutive integers. Then our average might be an integer, if our list is 1, 2, 3, or might not be, if our list is 1, 2, so this is not sufficient.
Using both Statements, we have an odd number of consecutive integers. With an odd number of things, the median must be in our set, so the median of this set must be equal to one of the integers in the set, and the median is an integer. But in a set of consecutive integers, the average equals the median, so the average is also an integer, and the two Statements together are sufficient. C.
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