6 — Factors, Multiples, and Number Theory

Factors, Multiples, and Number Theory

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Numbers have special relationships with each other, just like people do! Factors are numbers that divide evenly into another number, while multiples are what you get when you multiply a number by whole numbers. Understanding these relationships helps us solve real-world problems like sharing items equally or finding patterns in skip counting.

Tip

Key Rule: Factors divide INTO a number evenly (no remainder), while multiples come FROM multiplying a number by 1, 2, 3, 4, and so on.

Introduction to Factors, Multiples, and Number Theory · 1:39

Worked Example

Problem

Find all the factors of 12.

Solution

The factors of 12 are: 1, 2, 3, 4, 6, and 12

Explanation

  1. 1Start with 1 and the number itself (1 and 12 are always factors).
  2. 2Test if 2 divides evenly: 12 ÷ 2 = 6 ✓ (so both 2 and 6 are factors).
  3. 3Test if 3 divides evenly: 12 ÷ 3 = 4 ✓ (so both 3 and 4 are factors).
  4. 4Test if 5 divides evenly: 12 ÷ 5 = 2.4 ✗ (not a whole number, so 5 is not a factor).
  5. 5We already found all factors: 1, 2, 3, 4, 6, 12.

Worked Example

Problem

List the first 5 multiples of 7.

Solution

The first 5 multiples of 7 are: 7, 14, 21, 28, 35

Explanation

  1. 1Multiply 7 by 1: 7 × 1 = 7.
  2. 2Multiply 7 by 2: 7 × 2 = 14.
  3. 3Multiply 7 by 3: 7 × 3 = 21.
  4. 4Multiply 7 by 4: 7 × 4 = 28.
  5. 5Multiply 7 by 5: 7 × 5 = 35. Remember: Multiples go on forever, but we only need the first 5!

Tip

Memory Trick: Factors are 'friends' that fit perfectly into a number, while multiples are like a number's 'family tree' - they keep growing bigger!

Tip

Remember: Every number is both a factor of itself AND a multiple of itself. For example, 8 is a factor of 8 (because 8 ÷ 8 = 1) and 8 is also the first multiple of 8 (because 8 × 1 = 8).