"If a, b, and c Are Consecutive Positive Integers" — 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 a, b, and c are consecutive positive integers and a < b < c, which of the following must be true?
I. c − a = 2
II. a × b × c is an even integer
III. (a + b + c) / 3 is an integer
(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III
Write the Constraints
Before picking any numbers, write down what the problem tells us:
- a, b, c are consecutive
- a, b, c are positive
- a, b, c are integers
- a < b < c
Four constraints. Missing any one of them can send the whole problem off the rails. This is the most important step on a Roman numeral question, and the one that gets skipped most often.
Step 1: Pick a Simple Test Case
The smallest, easiest values that fit every constraint are 1, 2, 3.
Set up a simple column structure:
| a | b | c |
|---|---|---|
| 1 | 2 | 3 |
That is the visual organization system. Three columns. One row of values. Easy to extend if we need more tests.
Step 2: Check Roman Numeral I
I. c − a = 2
With our test values: 3 − 1 = 2. True.
If we try another set — say 2, 3, 4 — we get 4 − 2 = 2. True again.
The reason is straightforward once the testing surfaces it: consecutive integers are spaced by 1, so the third one is always 2 more than the first. Roman numeral I must be true.
Step 3: Check Roman Numeral II
II. a × b × c is an even integer.
With 1, 2, 3: 1 × 2 × 3 = 6. Even. True.
Try a different set. With 2, 3, 4: 2 × 3 × 4 = 24. Even. True.
What about 3, 4, 5? 3 × 4 × 5 = 60. Even. True.
The pattern: any three consecutive integers will include at least one even number. Multiplying any integer by an even integer gives an even result. Roman numeral II must be true.
Step 4: Check Roman Numeral III
III. (a + b + c) / 3 is an integer.
With 1, 2, 3: (1 + 2 + 3) / 3 = 6 / 3 = 2. Integer. True.
Try 2, 3, 4: (2 + 3 + 4) / 3 = 9 / 3 = 3. Integer. True.
Try 4, 5, 6: (4 + 5 + 6) / 3 = 15 / 3 = 5. Integer. True.
The middle number is always the average of three consecutive integers. So the sum divided by 3 always equals the middle number — which is an integer. Roman numeral III must be true.
All Three Must Be True
I, II, and III all hold across every test case.
The answer is (E).
Why This Problem Matters
About 14% of test takers miss this one. The math is light. The risk comes from the four stacked constraints — consecutive, positive, integer, and ordered. If you forget any of them, you can pick test values that lead you astray.
Two takeaways that scale to harder number properties questions:
The first is writing constraints before picking numbers. This sounds basic. It is also the thing that separates a 14% miss rate from a 37% miss rate (which is what happens on Problem 3, where the constraints stack even higher).
The second is testing more than one case. Roman numeral I happens to be true for every set of consecutive integers, but you would not necessarily know that from a single test. Trying a second case both confirms the pattern and often surfaces the theoretical reason for it.
This problem is the warm-up. The real test of the system is what happens when the constraints get harder to track. That is what the next two problems in the episode are designed to show.
Next problem: If the Sum of N Consecutive Integers Is 0 — GMAT® Worked Solution
Back to the strategy article: Testing Numbers on the GMAT®: A Simple System That Beats Memorizing Number Theory