Pencil’s up! Here is the answer to yesterday’s Challenge Question.
Question
If x and n are positive integers, is n = 1?
(1) The sum of n consecutive integers, starting at x, is divisible by xn.
(2) The product of n consecutive integers, starting at x, is divisible by x^n.
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Answer
The problem asks a “Yes or No” question about n: is it equal to 1? Note that we do not need to know the value of n in order to answer this question definitively. For instance, knowing that n is even would be sufficient to answer the question “No” (which would be a sufficient answer).
Statement (1) asserts that the sum of n consecutive integers, starting at x, is divisible by xn. First, we can connect this with other facts about consecutive integers. It turns out that the sum of n consecutive integers is divisible by n if and only if n is odd. (The reason is that the average number in a set of consecutive integers is actually the middle integer if you have 3, 5, or some other odd number of integers. But if you have an even number of integers, there is no middle integer, and the average number is not an integer. This matters because the sum of n consecutive integers divided by n IS the average number in a set of consecutive integers.) So we rule out even values of n. However, if we simply let x be 1, then the condition is satisfied by n = 1, 3, 5, or any other positive odd integer. We do not know whether n is equal to 1. INSUFFICIENT.
Statement (2) seems even more complicated, but it can be defeated again by a judicious choice of x as 1. If x = 1, then x^n = 1, no matter what n is. Since all positive integers are divisible by 1, then there is no restriction on the value of n, which could be equal to 1 OR to any other positive integer. INSUFFICIENT.
Statements (1) and (2) together: Putting what we have learned together, we know that if we let x = 1, then n might equal 1, but it could also equal any other positive odd number. We cannot answer the question definitively. INSUFFICIENT. The correct answer is (E): Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
