"If the Sum of N Consecutive Integers Is 0" — GMAT® Worked Solution
From Episode 38 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the full strategy behind number properties questions on the GMAT®, read: Testing Numbers on the GMAT®: A Simple System That Beats Memorizing Number Theory.
The Problem
Source: Official Guide for GMAT® Review, 11th Edition
If the sum of N consecutive integers is 0, which of the following must be true?
I. N is an even number
II. N is an odd number
III. The average of the N integers is 0
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
Write the Constraints
Pull every piece of given information out of the problem before doing any math:
- The values are integers
- They are consecutive
- They sum to 0
- N is the count of those integers
This is a Roman numeral question, so getting all the constraints down early matters even more than usual. One missed constraint can flip your answer.
Step 1: Pick a Simple Test Case
We need consecutive integers that sum to 0. The simplest set: −1, 0, 1.
| a | b | c |
|---|---|---|
| −1 | 0 | 1 |
Track N alongside the set: N = 3.
Step 2: Check Roman Numeral I
I. N is an even number.
In our test case, N = 3. That is odd. So Roman numeral I is not true here.
That kills any answer choice that includes I — eliminate (A) and (D).
Step 3: Check Roman Numeral III
III. The average of the N integers is 0.
In our test: (−1 + 0 + 1) / 3 = 0 / 3 = 0. True.
This is actually a place where a piece of theory comes for free. The average of any set of numbers is the sum divided by the count. The problem tells us the sum is 0. So the average is 0 divided by something — which is always 0. Roman numeral III must be true.
That eliminates any answer choice without III. We are down to (D) or (E), and we already eliminated (D). So the answer is (E).
But let us still test Roman numeral II to make sure.
Step 4: Check Roman Numeral II
II. N is an odd number.
Our first test gave N = 3, which is odd. So far so good. But we need to try to find a counterexample — a set of consecutive integers that sums to 0 with an even N.
Try N = 4: consecutive integers around 0 would be something like −2, −1, 0, 1. Sum = −2. Not 0.
Try shifting: −1, 0, 1, 2. Sum = 2. Not 0.
Any time we try four consecutive integers, the sum will not be 0. That makes sense — with an even count, the values cannot be symmetric around 0 while staying consecutive integers.
Try N = 2: any two consecutive integers, like 0 and 1, sum to 1. Like −1 and 0, sum to −1. Never 0.
The pattern: when N is even, you cannot have consecutive integers that sum to 0. So N being odd is required.
Roman numeral II must be true.
All Together
II and III must be true. I is not required.
The answer is (E).
Why This Problem Matters
About 27% of test takers miss this one — nearly double the miss rate of Problem 1. The math is not harder. The trap is that the constraints are easier to lose track of.
The most common wrong answer is (C) — III only. Around 18% of test takers pick it. That tells a clear story: those test takers spotted that the average had to be 0 (the easy one), but did not work through whether N had to be odd.
What likely happened: they figured out III in their head, did not write much down, and skipped a thorough check of II under time pressure.
This is exactly the failure mode the testing system is designed to prevent. Three Roman numerals, three explicit checks. Each one written down. No reasoning shortcuts that skip a step.
One more thing to notice: the theoretical insight on III came for free from the testing setup. We did not have to memorize a rule about averages. The structure of the problem made the rule visible once we had a concrete example in front of us. That is part of the value of leading with testing — the theory you need often surfaces on its own.
Previous problem: If a, b, and c Are Consecutive Positive Integers
Next problem: If N = 4p, Where p Is a Prime Number Greater Than 2
Back to the strategy article: Testing Numbers on the GMAT®: A Simple System That Beats Memorizing Number Theory