GRE Math: Number Theory and Properties You Must Know

Number theory is a cornerstone of GRE Quantitative Reasoning. You won't see a section labeled "Number Theory," but the concepts—divisibility, factors, multiples, remainders, odds and evens—permeate every problem. Mastering GRE number theory gives you a massive advantage. This guide covers the essential properties, rules, and strategies you need to answer these questions quickly and accurately.

Why Number Theory Matters on the GRE

The GRE General Test is section-adaptive: your performance on the first Quantitative section determines the difficulty of the second. Number properties questions appear in both sections, and they often serve as a gateway to higher-level problems. If you're aiming for a top score, you can't afford to skip them.

These questions test your ability to reason with integers, recognize patterns, and apply rules under time pressure. The good news? Number theory is finite. Once you learn the core rules, you can apply them repeatedly. Let's break down the must-know concepts.

Core GRE Number Properties

Before diving into divisibility, ensure you have a firm grasp of the basic properties of integers. These form the foundation of every number theory question on the GRE.

Odd and Even Numbers

Parity rules are simple but powerful. Memorize them:

• Even ± Even = Even
• Odd ± Odd = Even
• Even ± Odd = Odd
• Even × Even = Even
• Even × Odd = Even
• Odd × Odd = Odd

These rules apply to addition, subtraction, and multiplication. Division is trickier: an even number divided by an even number can be even, odd, or not an integer at all.

Positive and Negative Numbers

Sign rules are equally important, especially when combined with parity. Remember:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative

When you see inequalities with variables, always consider negative values. The GRE loves to test whether you assume variables are positive.

Prime Numbers

A prime number is an integer greater than 1 with exactly two distinct positive factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23... Note that 1 is not prime, and 2 is the only even prime. The GRE expects you to know primes up to at least 50.

Prime factorization is a critical skill. Every integer greater than 1 can be expressed as a unique product of primes. For example, 60 = 2² × 3 × 5. This factorization is the key to finding factors, multiples, and greatest common divisors.

GRE Divisibility Rules

Divisibility questions are a staple of GRE number properties. You need to know the rules for common divisors, but more importantly, you need to apply them logically.

Basic Divisibility Rules

• 2: Last digit is even (0, 2, 4, 6, 8).
• 3: Sum of digits is divisible by 3.
• 4: Last two digits form a number divisible by 4.
• 5: Last digit is 0 or 5.
• 6: Divisible by both 2 and 3.
• 8: Last three digits form a number divisible by 8.
• 9: Sum of digits is divisible by 9.
• 10: Last digit is 0.

These GRE divisibility rules save time. Instead of dividing, you can quickly check if a number is a multiple. For example, is 1,234 divisible by 3? Sum the digits: 1+2+3+4=10, not divisible by 3. So 1,234 is not divisible by 3.

Divisibility by 7 and 11

The GRE rarely tests divisibility by 7 or 11 directly, but knowing the rules can help. For 11, alternate sum of digits: (sum of digits in odd positions) − (sum of digits in even positions) must be divisible by 11. For 7, double the last digit and subtract from the rest; if the result is divisible by 7, the original number is too.

Factors and Multiples

A factor divides evenly into a number; a multiple is the product of that number and an integer. The GRE often asks for the number of factors of an integer. To find it, use prime factorization: if n = p^a × q^b, the total number of factors is (a+1)(b+1). For example, 60 = 2² × 3¹ × 5¹, so it has (2+1)(1+1)(1+1) = 12 factors.

Multiples are straightforward: the multiples of 5 are 5, 10, 15, 20, and so on. The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. The greatest common divisor (GCD) is the largest number that divides both. These are often tested together.

Remainders and Modular Arithmetic

Remainder problems are common on the GRE. When you divide an integer n by a positive integer d, you get a quotient q and a remainder r, such that n = dq + r, where 0 ≤ r < d. This simple equation is the key to many problems.

For example, if n divided by 7 leaves a remainder of 3, then n = 7q + 3. You can then test possible values for n (3, 10, 17, 24...). The GRE often asks for the remainder when a large number is divided by a small number. Use patterns or modular arithmetic to solve these efficiently.

Modular arithmetic properties: (a + b) mod m = ((a mod m) + (b mod m)) mod m, and similarly for multiplication. This allows you to break down complex expressions.

Advanced Number Theory Concepts

For a top GRE Quant score, you need to go beyond the basics. These advanced topics appear in harder questions.

Least Common Multiple and Greatest Common Divisor

The LCM and GCD are related by the formula: LCM(a, b) × GCD(a, b) = a × b. This is useful when you know one value and need the other. To find GCD and LCM using prime factorization, take the lowest powers for GCD and highest powers for LCM. For example, 12 = 2² × 3 and 18 = 2 × 3². GCD = 2¹ × 3¹ = 6; LCM = 2² × 3² = 36.

Prime Factorization and Factor Counting

As mentioned, the number of factors of n = p^a × q^b is (a+1)(b+1). This extends to more primes. Also, the sum of factors can be found using the formula: (p^a+1−1)/(p−1) × ... but this is rarely tested. Focus on counting factors and finding specific factors.

Consecutive Integers

Consecutive integers have special properties. The product of n consecutive integers is always divisible by n!. For example, the product of any 3 consecutive integers is divisible by 6. Also, among any 2 consecutive integers, one is even. These facts are frequently tested in data sufficiency and quantitative comparison questions.

Units Digit Patterns

The units digit of powers cycles in a pattern. For example, powers of 3: 3, 9, 7, 1, 3, 9, 7, 1... The cycle length is 4. To find the units digit of 3⁴⁷, divide 47 by 4: remainder 3, so the units digit is the third in the cycle: 7. This is a classic GRE trick.

Common GRE Number Theory Traps

The GRE writers know where students slip up. Avoid these pitfalls:

• Assuming variables are positive integers. Always consider zero, negative numbers, and fractions unless stated otherwise.
• Forgetting that 1 is not prime. This is a common error.
• Misapplying divisibility rules. For example, divisibility by 6 requires both 2 and 3, not just one.
• Overlooking the remainder when the divisor is larger than the dividend. For example, 5 divided by 7 leaves a remainder of 5.
• Confusing factors and multiples. Factors are smaller or equal; multiples are larger or equal.

How to Practice GRE Number Theory

Mastery comes from targeted practice. Here's a plan:

• Learn the rules. Memorize divisibility rules, parity rules, and prime numbers up to 50.
• Apply to problems. Use official GRE practice questions and reputable prep materials.
• Review mistakes. Understand why you missed a question—was it a rule you forgot or a trap you fell into?
• Time yourself. Number theory questions can be solved quickly with the right approach. Aim for under 2 minutes per question.

For a structured approach, see our 2-month GRE study plan for working professionals. It breaks down topics week by week, including number theory.

For a schedule that adjusts to your available hours, a free GRE General Test study plan can help you stay organized across the weeks leading up to test day.

If you're also working on statistics, check out our guide to GRE Math Statistics: Mean, Median, and Standard Deviation. And for verbal, don't miss GRE Verbal Short Passage Tactics for Quick Points.

Integrating Number Theory with Other Topics

Number theory doesn't exist in isolation. It intersects with algebra, geometry, and data interpretation. For example, you might need to find the number of integer solutions to an inequality, or the area of a rectangle with integer sides and a given perimeter. Strong number sense makes these problems easier.

Also, remember that the GRE is adaptive. A strong performance on the first Quantitative section unlocks harder questions in the second. Number theory is often used to distinguish between average and high scorers. If you want a top score, you must be comfortable with these concepts.

To understand how your Quant score fits into the bigger picture, read What GRE Score Do You Need for Top Graduate Programs?.

Final Tips for GRE Number Theory

Here are your key takeaways:

• Master the basics: parity, sign, primes, factors, multiples.
• Memorize divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10.
• Use prime factorization for factors, GCD, and LCM.
• Practice remainder problems and units digit patterns.
• Always consider negative numbers and zero when variables are involved.
• Time yourself and review mistakes.

With consistent practice, number theory becomes a strength, not a weakness. It's one of the most predictable areas of GRE Quant, and mastering it can significantly boost your score.

To identify the number theory concepts that still feel shaky, take a free GRE General Test readiness check that offers practice questions with explanations and highlights your weakest topic.

Invest a few hours in GRE practice test now and save yourself a retake later.