Divisibility, Factors & Multiples
Divisibility is the machinery behind a surprising number of GRE questions — not just "is this divisible by 3?" but fractions that must simplify, ratios that must reduce, and word problems about events that line up on a schedule. A factor divides a number evenly; a multiple is what you get by multiplying it. The two ideas that pay off most are the greatest common divisor (GCD, also called HCF) and the least common multiple (LCM): GCD is about splitting things into equal groups, LCM is about when repeating cycles coincide. Learn the quick divisibility tests, and you will do in your head what the on-screen basic calculator would make slow and error-prone.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Digit-sum tests are your fastest friends: divisible by 3 (or 9) iff the digit sum is. The GRE calculator can never beat this.
- For a composite divisor, split into coprime parts: by 12 = by 3 AND by 4; by 15 = by 3 AND by 5.
- GCD = shared primes at the LOWEST power; LCM = all primes at the HIGHEST power. Say it that way and you never mix them up.
- GCD × LCM = a × b. Know any three of {a, b, GCD, LCM} and the fourth is one division away.
- Coprime (GCD = 1) ⇒ LCM = a × b. Spot coprimality and skip the factorisation.
- "Largest equal groups / biggest tile" → GCD. "When do cycles coincide / smallest common amount" → LCM.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Swapping GCD and LCM — taking highest powers for the GCD or lowest for the LCM.
- Applying GCD × LCM = product to THREE numbers. The identity holds for two numbers only.
- Testing divisibility by 4 using the whole number instead of just the last two digits (same for 8 → last three).
- Forgetting that a perfect square (like 36) has an odd number of factors because one pair repeats.
- Reading "factor" as "multiple". 4 is a factor of 12; 12 is a multiple of 4 — opposite directions.
- Assuming two even numbers are coprime. They always share the factor 2, so GCD ≥ 2.
📈 GRE exam insight & question patterns
Quantitative Comparison — Quantity A: GCD(a, b). Quantity B: LCM(a, b), for positive integers a, b. Which is greater?
Quantity B. The GCD never exceeds the smaller number and the LCM is never smaller than the larger, so LCM ≥ GCD always (equal only when a = b).
A scheduling problem: Bus X every 20 min, Bus Y every 30 min, both leave at 8:00. When do they next leave together?
LCM(20, 30) = 60 min, so 9:00. Any "meet/leave together again" cue points to LCM.
Numeric Entry — a rectangle 90 cm by 120 cm is tiled by identical square tiles with whole-cm sides, using the fewest tiles. Enter the tile side.
Fewest tiles means the LARGEST square, i.e. GCD(90, 120) = 30 cm.
Multiple-answer — which of 2, 3, 4, 6, 9 must divide any integer divisible by 36? Select all.
All of them. 36 = 2²·3², so every factor of 36 — including 2, 3, 4, 6 and 9 — divides any multiple of 36.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
- Divisibility tests (2, 3, 4, 5, 6, 9) replace slow calculator division with mental checks.
- Factors are finite and paired; multiples are infinite; the two words point in opposite directions.
- GCD = shared primes at lowest powers; it answers "largest equal group / biggest tile".
- LCM = all primes at highest powers; it answers "when do cycles coincide / smallest common amount".
- GCD × LCM = a × b for two numbers — find any one of the four from the other three.
- Coprime numbers have GCD 1, so their LCM is just the product.
- Split composite divisibility into coprime parts (12 = 3 and 4).
- On GRE word problems, "equal groups" signals GCD and "meet again" signals LCM.
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Divisibility, Factors & Multiples when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (5 topics) | 5/5 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 8 cards |
Formula Reference Sheet
Divisibility tests
| By 2 | last digit even |
|---|---|
| By 3 | digit sum divisible by 3 |
| By 4 | last two digits divisible by 4 |
| By 5 | ends in 0 or 5 |
| By 9 | digit sum divisible by 9 |
GCD & LCM
| GCD × LCM | GCD(a,b) × LCM(a,b) = a × b |
|---|---|
| LCM from GCD | LCM = (a × b) / GCD |
| Coprime numbers | GCD = 1 ⇒ LCM = a × b |
| Divisibility by 6 | divisible by 2 AND by 3 |