What This Episode Covers
Episode 37 of Real GMAT® Problems takes a deep dive into divisibility — one of those GMAT® quant topics that shows up constantly but rarely gets its own dedicated practice session. The episode works through three Official Guide problems that range from a quick warm-up to a multi-step elimination challenge, and uses them to introduce divisibility shortcuts that can save significant time on test day.
The broader theme is the same one that runs through the entire Real GMAT® Problems series: good habits and consistent execution matter more than raw talent. Divisibility questions are a perfect example — they're not conceptually difficult for most test-takers, but they have surprisingly high miss rates because people rush, skip writing things down, or abandon their process under time pressure.
Along the way, we cover specific divisibility rules for 3, 4, 5, 8, and 9, when to use them versus when to fall back on long division, and why good visual organization on your scratch work is almost always the difference between getting these right and making a careless mistake.
Problems Covered
Problem 1 — Warm-Up: (100 + N) / N For which of the following values of N is (100 + N) / N not an integer? Options: 1, 2, 3, 4, 5. We walk through plugging each answer in with clear visual organization — writing a big letter for each answer choice and working left to right. The divisibility-by-3 shortcut appears here: if the sum of the digits of a number equals a multiple of 3, the number is divisible by 3. Since 1 + 0 + 3 = 4, and 4 is not a multiple of 3, we know 103 is not divisible by 3. Answer: C.
Problem 2 — Mid-Level: N × (N+1) × (N+2) If N is a positive integer, then N × (N+1) × (N+2) is: (A) even only when N is even, (B) even only when N is odd, (C) odd whenever N is odd, (D) divisible by 3 only when N is odd, (E) divisible by 4 whenever N is even. This one has a 20% miss rate. The episode demonstrates why plugging in small numbers with good scratch work beats trying to reason through each answer theoretically. Testing N = 1 eliminates A and C. Testing N = 2 eliminates B and D. Answer: E. The main trap is that options A and D are designed to catch people who use theoretical reasoning too quickly and commit to an answer without reading it carefully.
Problem 3 — Harder: Lowest Positive Integer Divisible by 2–9 Which of the following is the lowest positive integer divisible by 2, 3, 4, 5, 6, 7, 8, and 9? Options: 15,120 / 3,024 / 2,520 / 1,890 / 1,680. This has a 26% miss rate — and 11% of test-takers go for option A (which is divisible by everything but is not the lowest). We use divisibility shortcuts to eliminate options one by one: the divisibility-by-5 rule knocks out B, the divisibility-by-4 rule knocks out D, the divisibility-by-9 rule knocks out E. Then long division by 7 and 8 on the remaining two options confirms C (2,520) as the answer.
Key Takeaways
- Divisibility shortcuts are worth memorizing. Rules for 3, 4, 5, and 9 come up frequently on the GMAT® and can replace multi-step long division in many cases. The time savings compound across a section.
- The divisibility-by-3 rule: sum the digits. If the sum is a multiple of 3, the number is divisible by 3. Same logic applies to 9 — if the digit sum is a multiple of 9, the number is divisible by 9.
- The divisibility-by-4 rule: check the last two digits. If the number formed by the last two digits is divisible by 4, the whole number is divisible by 4.
- Good visual organization prevents careless errors. Write a big letter for each answer choice. Give each one its own row or column. This takes seconds and prevents the most common reason people miss questions they know how to do.
- Read the full answer choice before committing. On Problem 2, 12% of test-takers pick D because they recognize that three consecutive integers are always divisible by 3 — and stop reading before they see "only when N is odd." Same trap on Problem 3 — 11% pick A because it is divisible by everything, but the question asks for the lowest.
- Long division is still worth practicing. Shortcuts handle many cases, but some divisors (like 7) do not have reliable shortcuts. Comfortable long division is a safety net that always works.
Related Reading
- Real GMAT® Problems - Ep. 38 - The Power of Testing Numbers — builds directly on this episode's plugging-in-answers framework with number properties problems
- Real GMAT® Problems - Ep. 35 - Plugging In Answers — covers the plugging-in-answers technique referenced throughout this episode