Integers & Number Properties
Number properties are the grammar of GRE Quant. The section almost never asks you to compute something hard; instead it asks whether a statement about integers is always true, and the winners are the people who know the rules of signs, parity (odd/even), and the peculiar behaviour of 0 and 1 cold. This chapter is the one place where memorising a short list of rules pays off on nearly every test, because those rules turn slow arithmetic into instant reasoning. Master them and you can answer many Quantitative Comparison questions without computing a single value — you simply decide what the sign or the parity of an expression must be.
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.
- On "is it always true?" questions, test small special cases in this order: 0, 1, −1, a fraction, a large number. One of them usually breaks a false claim.
- Parity beats arithmetic: to check an answer, compute the odd/even of an expression instead of the value — it is faster and catches sign slips.
- A product of n consecutive integers is divisible by n! (2 in a row → by 2, 3 in a row → by 6, 4 in a row → by 24).
- To reverse an inequality you must multiply or divide by a negative — the on-screen calculator will not warn you, so flag the flip yourself.
- |x − c| is "distance from c". Turn every absolute-value inequality into a distance statement and read the range off the number line.
- When a denominator or divisor is a variable, immediately ask "can it be 0?" — that single check resolves many Quantitative Comparison traps.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Assuming a "number" is a positive integer. Unless told so, it may be negative, zero, or a fraction — the source of most "cannot be determined" answers.
- Writing −3² = 9. Without parentheses the exponent binds first: −3² = −9, while (−3)² = 9.
- Forgetting to flip the inequality sign when multiplying or dividing by a negative.
- Solving |x| = a with only one answer. It gives two: x = a and x = −a (when a > 0).
- Calling 1 prime, or forgetting that 0 is even. Both cost easy points on definition questions.
- Treating |a + b| as |a| + |b|. They are equal only when a and b have the same sign.
📈 GRE exam insight & question patterns
Quantitative Comparison — x is an integer. Quantity A: x². Quantity B: x. Which is greater?
Cannot be determined. For x = 2, A > B; for x = 1, they are equal; for x = 0, equal; the fractional/negative cases the GRE wants you to test change the outcome, so no single relationship holds.
A question states "n is a positive even integer". Why does that wording matter?
It removes 0, negatives and fractions from play, so parity rules apply cleanly. The GRE adds or omits each of those three words on purpose — read them like part of the equation.
Numeric Entry — the product of three consecutive integers is 210. Enter the largest of the three.
5 × 6 × 7 = 210, so the integers are 5, 6, 7 and the largest is 7. (Use divisibility by 6 to narrow the search quickly.)
Multiple-answer — which of these must be even for every integer n? Select all: n²+n, 2n+1, n(n−1), 3n.
n²+n = n(n+1) is even, and n(n−1) is even (both are products of consecutive integers). 2n+1 is always odd; 3n has the parity of n. So the even-for-all choices are n²+n and n(n−1).
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
- Integers are whole numbers; watch for the word "integer" vs "number" — it decides whether fractions and negatives are allowed.
- Parity rules (odd/even) let you answer many questions without computing a value.
- Signs: same-sign products are positive, an odd count of negatives is negative, and negative bases flip sign only on odd powers.
- Multiplying or dividing an inequality by a negative reverses it.
- |x| is distance from 0: |x| = a branches into two cases, and |x − c| reads as a range on the number line.
- 0 is even and neither positive nor negative; division by 0 is undefined; x⁰ = 1 for x ≠ 0; 1 is not prime.
- Products of consecutive integers are highly divisible — n in a row is divisible by n!.
- On Quantitative Comparison, test 0, 1, −1 and a fraction before deciding; "cannot be determined" is a real answer.
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Integers & Number Properties 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 | 10 cards |
Formula Reference Sheet
Parity (odd / even)
| even ± even | even |
|---|---|
| odd ± odd | even |
| odd ± even | odd |
| even × anything | even |
| odd × odd | odd |
Signs & special numbers
| negative × negative | positive |
|---|---|
| (negative)^even | positive |
| (negative)^odd | negative |
| Absolute value | |x| = distance from 0 (never negative) |
| n consecutive integers | product divisible by n! |