Number Properties

What This Module Covers

This module covers the foundational number theory concepts that appear throughout the GRE Quantitative section: odd/even rules, prime numbers and prime factorization, LCM and GCF, divisibility rules, remainders, units digit patterns, and consecutive integers. You'll see these concepts in Problem Solving, Quantitative Comparison, and Data Interpretation questions — often layered together in a single problem.

Why It Matters on the GRE

Number properties questions appear in almost every GRE Quant section, and mastering them unlocks faster, more confident answers across algebra, arithmetic, and word problems. Students who skip this foundation consistently lose points on questions that look computational but are actually conceptual.

Core Concepts

Odd and Even Rules

An integer is even if it's divisible by 2, and odd if it's not. Zero is even. Negatives follow the same rules: −4 is even, −7 is odd.

For addition and subtraction, the rules are:

• odd + odd = even
• even + even = even
• odd + even = odd

For multiplication, the rule is simpler: any even factor makes the entire product even. The only way a product stays odd is if every single factor is odd.

Must Know: odd × odd = odd, but anything × even = even — one even factor contaminates the whole product.

Example: If a = 3 and b = 5 (both odd), what is 2a + b?

1. 2a = 2 × 3 = 6 (even × odd = even)
2. 6 + b = 6 + 5 = even + odd = odd

You don't need to calculate — the parity rules get you to "odd" without arithmetic.

Prime Numbers

A prime number is a positive integer with exactly two distinct factors: 1 and itself. That definition is precise and testable — memorize it word for word.

The first eight primes are: 2, 3, 5, 7, 11, 13, 17, 19. Know these cold. Two critical edge cases trip up almost every test-taker:

Must Know: 1 is NOT prime (only one factor: itself). 2 IS prime — and it's the only even prime.

The GRE loves to ask questions where the correct answer hinges entirely on whether you remember that 1 is not prime or that 2 is even but still prime. When a question says "p is prime," always ask yourself: could p = 2?

Prime Factorization

Every integer greater than 1 can be broken down into a unique product of prime numbers. This is called the prime factorization, and it's the engine behind LCM, GCF, and divisibility work.

The method: divide by the smallest prime that goes in evenly, then repeat on the quotient until you're left with 1.

Example: Find the prime factorization of 60.

1. 60 ÷ 2 = 30
2. 30 ÷ 2 = 15
3. 15 ÷ 3 = 5
4. 5 is prime — stop.

Result: 60 = 2² × 3 × 5

Must Know: Prime factorization is unique. There is exactly one way to express any integer as a product of primes (ignoring order). This is the Fundamental Theorem of Arithmetic.

GCF and LCM

The Greatest Common Factor (GCF) of two numbers is the largest integer that divides both evenly. The Least Common Multiple (LCM) is the smallest positive integer that both numbers divide into evenly.

Use prime factorization to find both:

• GCF: take the minimum exponent of each shared prime.
• LCM: take the maximum exponent of every prime that appears in either number.

Example: Find GCF and LCM of 36 and 48.

1. 36 = 2² × 3²
2. 48 = 2⁴ × 3
3. GCF = 2² × 3¹ = 4 × 3 = 12 (minimum exponents of shared primes)
4. LCM = 2⁴ × 3² = 16 × 9 = 144 (maximum exponents of all primes)

Must Know: LCM × GCF = the product of the two original numbers. So LCM × GCF = 36 × 48 = 1,728. Check: 144 × 12 = 1,728. ✓

Divisibility Rules

Checking divisibility without long division saves real time on the GRE. Know these cold:

Example: Is 5,814 divisible by 6?

1. Check 2: last digit is 4 (even) → yes
2. Check 3: 5 + 8 + 1 + 4 = 18, and 18 ÷ 3 = 6 → yes
3. Divisible by both 2 and 3 → divisible by 6 ✓

Remainders

When you divide integer a by integer d, the relationship is: dividend = divisor × quotient + remainder, or a = d × q + r, where 0 ≤ r < d.

The GRE frequently uses remainder cycles — the remainders when dividing successive integers by a fixed number always repeat in a predictable pattern. This is the key to "last digit" questions and many pattern problems.

Must Know: When n leaves remainder r after dividing by d, then n + d also leaves remainder r. Remainders cycle with period d.

Example: What is the remainder when 2²⁰ is divided by 7?

1. Check powers of 2 mod 7: 2¹=2, 2²=4, 2³=8→1, 2⁴=16→2, 2⁵=32→4, 2⁶=64→1 ...
2. The cycle (mod 7) is: 2, 4, 1, 2, 4, 1 — period 3.
3. 20 ÷ 3 = 6 remainder 2. So 2²⁰ has the same remainder as 2² = 4.
4. Remainder = 4

Units Digit Patterns

The units digit of a power depends only on the units digit of the base — and the pattern always repeats with a cycle length of 1, 2, or 4. You only need to track the last digit through each multiplication.

Key cycles to memorize:

Must Know: For cycle-length-4 bases, divide the exponent by 4. The remainder tells you the position in the cycle (remainder 0 → use position 4).

Example: What is the units digit of 7⁴³?

1. Cycle for 7: 7, 9, 3, 1 (length 4)
2. 43 ÷ 4 = 10 remainder 3
3. Position 3 in the cycle = 3

Consecutive Integers

Consecutive integers are written n, n+1, n+2, ... Consecutive even or odd integers are written n, n+2, n+4, ... (where n is even or odd respectively).

Must Know: Any set of n consecutive integers contains exactly one multiple of n. This means the product of any n consecutive integers is always divisible by n!.

This is why the product of any 2 consecutive integers is even, the product of any 3 consecutive is divisible by 6, and so on.

Common Traps

• Assuming variables are positive: The GRE often hides zero or negative numbers in variable problems. If a problem says "n is an integer," zero and negative values are fair game unless specifically excluded.
• Forgetting 0 is even: Zero has no parity confusion on the GRE — it is definitively even. It's also a multiple of every integer.
• Confusing factors and multiples: Factors of 12 are 1, 2, 3, 4, 6, 12 (smaller or equal). Multiples of 12 are 12, 24, 36, ... (larger or equal). Factors ≤ the number; multiples ≥ the number.
• Thinking 1 is prime: 1 has only one factor (itself), so it fails the "exactly two factors" test. It is not prime and not composite.
• Forgetting 2 is prime: 2 is prime, and it's the only even prime. When a question restricts variables to primes, 2 is almost always the key test case.

GRE Strategy

• On any variable problem involving primes, immediately test p = 2 as your first case. It breaks more assumptions than any other prime.
• When a problem asks for the units digit of a large power, do NOT compute the full power. Find the cycle, find the remainder, done.
• LCM and GCF problems are almost always faster with prime factorization than with listing factors or multiples.
• "How many factors does n have?" → Use the prime factorization: n = p^a × q^b × r^c → number of factors = (a+1)(b+1)(c+1).
• On Quantitative Comparisons involving divisibility or remainders, plug in 0 and negatives early — they often flip the comparison.

Worked Example

Question:

Quantity A: The number of distinct prime factors of 120 Quantity B: The number of distinct prime factors of 90

(A) Quantity A is greater. (B) Quantity B is greater. (C) The two quantities are equal. (D) The relationship cannot be determined from the information given.

Solution:

Step 1: Prime factorize 120.

120 ÷ 2 = 60 60 ÷ 2 = 30 30 ÷ 2 = 15 15 ÷ 3 = 5 5 is prime.

120 = 2³ × 3 × 5

Distinct prime factors of 120: {2, 3, 5} → 3 distinct primes

Step 2: Prime factorize 90.

90 ÷ 2 = 45 45 ÷ 3 = 15 15 ÷ 3 = 5 5 is prime.

90 = 2 × 3² × 5

Distinct prime factors of 90: {2, 3, 5} → 3 distinct primes

Step 3: Compare.

Quantity A = 3, Quantity B = 3. They are equal.

Answer: (C)

Why the wrong answers fail:

• (A) and (B) are wrong because students often confuse "number of distinct prime factors" with "total prime factors counting multiplicity." If the question asked for the sum of exponents, 120 would give 3+1+1=5 and 90 would give 1+2+1=4, favoring Quantity A. But "distinct" means we only care about the unique primes, not how many times each appears. Both numbers share the exact same set of prime bases: 2, 3, and 5.

• (D) is wrong because both quantities are fixed integers — there are no variables, so the relationship is fully determined. Only use (D) when a variable could take multiple values that flip the comparison.

Verification: List the distinct primes for each. 120 = 2³ × 3 × 5 → primes are 2, 3, 5. 90 = 2 × 3² × 5 → primes are 2, 3, 5. Same set, same count. ✓