Primes & Prime Factorization
Prime factorization is the single most powerful tool in GRE arithmetic. Every integer greater than 1 breaks into primes in exactly one way — that is the Fundamental Theorem of Arithmetic — and once you have that breakdown, questions about factors, perfect squares, GCD and LCM all become bookkeeping on the exponents. A prime has exactly two positive factors, 1 and itself; a composite has more. The non-obvious payoff is the factor-counting rule: you never list divisors on the GRE, you read their count off the prime exponents. This chapter builds that habit, because the test rewards students who see 720 not as a number to divide but as 2⁴·3²·5.
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.
- Factor once, answer everything: from n = p^a·q^b·r^c you get the divisor count (a+1)(b+1)(c+1), GCD, LCM, square/cube tests — all from the exponents.
- A perfect square ⇔ an odd number of factors ⇔ all prime exponents even. Three views of the same fact.
- To test primality, only try prime divisors up to √n. For 97 you stop at 7.
- To make a number a perfect square, top up each odd exponent to the next even; for a cube, top up to the next multiple of 3.
- Perfect squares end only in 0, 1, 4, 5, 6, 9. A number ending in 2, 3, 7 or 8 is never a perfect square.
- Memorise squares to 30 and cubes to 10 — they let you place roots between benchmarks without the calculator.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Calling 1 prime, or forgetting 2 is the only even prime.
- Listing factors by hand instead of using (a+1)(b+1)(c+1) — slow and error-prone.
- Counting distinct primes when the question asks for the number of factors (or vice versa).
- Adding exponents when multiplying the divisor-count terms, instead of multiplying them.
- Thinking an even exponent is needed for a cube — cubes need exponents divisible by 3, not by 2.
- Assuming √n is irrational without checking: many GRE radicands are perfect squares in disguise.
📈 GRE exam insight & question patterns
Quantitative Comparison — Quantity A: the number of positive factors of 2⁴·3². Quantity B: the number of positive factors of 2·3·5·7. Which is greater?
A: (4+1)(2+1) = 15. B: (1+1)⁴ = 16. Quantity B is greater — more distinct primes can beat higher exponents.
Numeric Entry — enter the smallest positive integer k such that 90k is a perfect square.
90 = 2·3²·5. The odd exponents are on 2 and 5, so k = 2·5 = 10 (giving 900 = 30²).
A number has exactly 3 positive factors. What can you conclude about it?
It is the square of a prime (like 4, 9, 25, 49). Only p² gives (2+1) = 3 factors.
Multiple-answer — which of 2, 3, 4, 9, 36 must divide n if n = 2²·3²·k for a positive integer k? Select all.
All of them: 2, 3, 4 = 2², 9 = 3² and 36 = 2²·3² all appear within 2²·3², so each divides n regardless of k.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
- Every integer > 1 has a unique prime factorization — the "DNA" every other question reads from.
- A prime has exactly two factors; 1 is not prime and 2 is the only even prime.
- Count factors with (a+1)(b+1)(c+1) from the exponents — never list them.
- An odd number of factors means the number is a perfect square.
- Perfect squares have all-even prime exponents; perfect cubes have exponents divisible by 3.
- To reach a square or cube, multiply by the primes that top up the deficient exponents.
- Test primality only up to √n, using prime divisors.
- Squares end only in 0,1,4,5,6,9 — a quick disqualifier under time pressure.
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Primes & Prime Factorization 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 (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 8 cards |
Formula Reference Sheet
Primes & factorization
| Prime | exactly two factors: 1 and itself |
|---|---|
| Smallest / only even prime | 2 |
| Fundamental Theorem | unique prime factorization for n > 1 |
| Primality check up to | √n (test primes ≤ √n) |
Counting from exponents
| n = p^a · q^b · r^c | number of factors = (a+1)(b+1)(c+1) |
|---|---|
| Perfect square | every prime exponent is even |
| Perfect cube | every prime exponent is a multiple of 3 |
| Odd number of factors ⇔ | n is a perfect square |