Scholars & Parents
ExplainerMathGrades 3–5

Factors and Multiples

What Is This?

Factors and multiples are two sides of the same coin — they both describe relationships between numbers through multiplication and division.

  • A factor of a number divides into it exactly, with no remainder.
  • A multiple of a number is what you get when you multiply it by a whole number.

Example: 4 is a factor of 12 (because 12 ÷ 4 = 3 exactly). 12 is a multiple of 4 (because 4 × 3 = 12).

Why Does It Matter?

Factors and multiples are the foundation of fractions (simplifying, finding common denominators), number theory (prime numbers, divisibility), and algebra. Every time you simplify a fraction or find an equivalent one, you are using factors and multiples.

Part 1 — Factors

What Is a Factor?

A factor of a number is any whole number that divides into it exactly.

f is a factor of n if n ÷ f has no remainder.

Finding All Factors

The most reliable method is to test factor pairs systematically, starting from 1.

Example: Find all factors of 24.

Factor pair Check
1 × 24 ✓
2 × 12 ✓
3 × 8 ✓
4 × 6 ✓
5 × ? 24 ÷ 5 = 4.8 — not a whole number ✗
6 × 4 Already found — stop here

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Tip: Stop testing when the factor you are trying exceeds the square root of the number. For 24, √24 ≈ 4.9, so you can stop after testing 4.

Factor Pairs

Every factor has a partner. Writing them in pairs makes sure you don't miss any:

  • 24 = 1 × 24 = 2 × 12 = 3 × 8 = 4 × 6

Every Number's Guaranteed Factors

Every whole number greater than 0 has at least two factors: 1 and itself. (The only exception is 1, which has just one factor: itself.)

Part 2 — Multiples

What Is a Multiple?

A multiple of a number is the result of multiplying it by any positive whole number.

Multiples of n: n × 1, n × 2, n × 3, n × 4, …

Multiples of 6: 6, 12, 18, 24, 30, 36, … (the list goes on forever)

Multiples of 9: 9, 18, 27, 36, 45, 54, …

Notice that the multiples of any number form the times table for that number.

Is a Number a Multiple of Another?

A number is a multiple of n if n is one of its factors.

  • Is 42 a multiple of 7? → 42 ÷ 7 = 6 exactly → Yes ✓
  • Is 50 a multiple of 8? → 50 ÷ 8 = 6.25 → No ✗

Part 3 — Greatest Common Factor (GCF)

The Greatest Common Factor (also called the Highest Common Factor, HCF) of two numbers is the largest factor they share.

Method — List and Compare

Example: GCF of 12 and 18

  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Factors of 18: 1, 2, 3, 6, 9, 18
  • Common factors: 1, 2, 3, 6
  • GCF = 6 ✓

Why GCF Matters

The GCF is used to simplify fractions to their lowest terms:

  • 12/18 ÷ 6/6 = 2/3 ✓

Part 4 — Least Common Multiple (LCM)

The Least Common Multiple of two numbers is the smallest multiple they share.

Method — List and Compare

Example: LCM of 4 and 6

  • Multiples of 4: 4, 8, 12, 16, 20, 24, …
  • Multiples of 6: 6, 12, 18, 24, …
  • First common multiple: 12
  • LCM = 12 ✓

Why LCM Matters

The LCM is used to find a common denominator when adding or subtracting fractions:

  • 1/4 + 1/6: common denominator = 12 → 3/12 + 2/12 = 5/12 ✓

Connecting Factors and Multiples

These four relationships are all true at the same time:

  • 4 is a factor of 24
  • 24 is a multiple of 4
  • 24 is divisible by 4
  • 4 divides 24

All four mean exactly the same thing.

Worked Examples

Worked Example 1: All factors of 36

  • 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
  • Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

Worked Example 2: GCF of 24 and 36

  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  • Common: 1, 2, 3, 4, 6, 12 → GCF = 12 ✓

Worked Example 3: LCM of 6 and 9

  • Multiples of 6: 6, 12, 18, 24
  • Multiples of 9: 9, 18, 27
  • First common: 18 → LCM = 18 ✓

Worked Example 4: LCM of 8 and 12

  • Multiples of 8: 8, 16, 24, 32
  • Multiples of 12: 12, 24, 36
  • First common: 24 → LCM = 24 ✓

Worked Example 5 (Word Problem):

Buses on Route A leave every 8 minutes. Buses on Route B leave every 12 minutes. They both leave at 9:00 am. When is the next time they leave together?

  • LCM of 8 and 12 = 24 minutes
  • Next simultaneous departure: 9:24 am ✓

Common Mistakes

Mistake 1: Confusing factors and multiples "6 is a factor of 12" (6 divides 12) vs "12 is a multiple of 6" (12 = 6 × 2). These are reverse statements of the same relationship.

Mistake 2: Missing factor pairs Always work systematically from 1 upward in pairs. Missing the middle factors (like 4 and 6 in factors of 24) is the most common error.

Mistake 3: Including non-factors 5 is NOT a factor of 24 because 24 ÷ 5 = 4.8 (not a whole number). Division must be exact.

Mistake 4: Confusing GCF and LCM GCF → smaller (greatest shared factor). LCM → larger (smallest shared multiple). For 4 and 6: GCF = 2, LCM = 12.

Key Takeaways

  • A factor of n divides into n exactly; every number has 1 and itself as factors
  • Find factors by testing pairs from 1 upward — stop when factors begin to repeat
  • A multiple of n is n × (any positive whole number) — the list is infinite
  • GCF = largest factor shared by two numbers (used to simplify fractions)
  • LCM = smallest multiple shared by two numbers (used to find common denominators)
  • Factors and multiples are inverse relationships: if a is a factor of b, then b is a multiple of a

Practice and Resources

Ready to practise? Try our Grade 3-5 Factors and Multiples Worksheet covering factor pairs, GCF, LCM, and word problems. Test yourself with the Grade 3-5 Math Practice Test.

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