"If N = 4p, Where p Is a Prime Number Greater Than 2" — 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 N = 4p, where p is a prime number greater than 2, how many different positive even divisors does N have, including N?
(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight
Write Every Constraint
This problem looks simpler than the first two, and the math is. The difficulty is in how many constraints there are to track.
Pull them all out before doing any math:
- N = 4p (the relationship between N and p)
- p is prime
- p is greater than 2
- We want divisors that are different (no repeats)
- We want divisors that are positive
- We want divisors that are even
- N counts as one of the divisors
Seven constraints. Working memory holds about seven items at a time. That is part of why this question has a 37% miss rate — there is barely enough room in your head to hold the question itself, let alone do the math.
Writing them down solves that problem.
Step 1: Pick a Value for p
p has only two constraints — prime, and greater than 2 — so it is the easier variable to plug in for.
The smallest prime greater than 2 is 3. Use p = 3.
Then N = 4 × 3 = 12.
Write that down explicitly: N = 12. This single line is what keeps the rest of the problem from spiraling.
Step 2: List the Divisors of 12 Using Factor Pairs
A divisor of 12 is any number that divides into 12 with an integer result. The cleanest way to list them is by factor pairs:
- 1 × 12
- 2 × 6
- 3 × 4
That covers every divisor of 12: 1, 2, 3, 4, 6, 12.
Step 3: Apply the Constraints
Now filter the list using the four constraints on what counts.
Different: no repeats in our list. All six are distinct.
Positive: all six are positive.
Even: cross off 1 and 3 (both odd). We are left with 2, 4, 6, 12.
Including N: 12 is in our list. Good.
Step 4: Count
Even divisors of 12: 2, 4, 6, 12.
That is four divisors.
The answer is (C).
A Note on Why One Test Is Enough Here
This is not a Roman numeral question. It is a single-answer question with numeric choices.
Because the answer to the question must hold for any valid value of p, the answer choice that works for our test case has to be the correct one. There is no risk of one value of p giving us 4 even divisors and another value giving us 6.
So we are done after one test. Quick verification with another prime — say p = 5, giving N = 20 — confirms it. Factor pairs of 20: 1 × 20, 2 × 10, 4 × 5. Even divisors: 2, 4, 10, 20. Still four.
Why This Problem Matters
About 37% of test takers miss this one. Almost 40% higher miss rate than Problem 2, and roughly 2.6 times the miss rate of Problem 1.
The math is the easiest of the three.
What makes it harder is the constraint count. Seven separate pieces of information to hold while you do the math. If you try to reason through this in your head — and a lot of test takers do, because the question looks short — you can easily drop one. Forget "different," and you count 2 × 2 = 4 as a separate factor. Forget "even," and you include 1, 3. Forget "including N," and you leave 12 off the list.
This is the case for organized scratch work even on questions that look simple. The same testing-numbers system we used on the first two problems applies here, with one change: because the question is not Roman numeral, we only need one test case.
A useful pattern to internalize: on single-answer numeric questions where the variables are constrained, one test case is enough. On Roman numeral questions, plan for at least two.
That distinction alone can save you a minute on questions like this one, while keeping your accuracy where you want it.
Previous problem: If the Sum of N Consecutive Integers Is 0
Back to the strategy article: Testing Numbers on the GMAT®: A Simple System That Beats Memorizing Number Theory