Factors Multiples

Factors & Multiples

What you'll learn

How factors (divisors) and multiples connect multiplication and division.
Systematic methods to list all factors, find common factors, and identify prime and composite numbers.
How prime factorisation leads to HCF and LCM — the two most tested ideas in Class 6 number theory.
Real-life uses such as scheduling events that repeat at different intervals.

Key concepts

Level 1 — Factors and multiples

Verbal: A factor of a number divides it exactly (no remainder). A multiple is what you get when you multiply that number by a whole number.

Symbolic: If 12 ÷ 3 = 4 exactly, then 3 and 4 are factors of 12, and 12 is a multiple of 3.

Visual (factor pairs for 24):

Pair	Check
1 × 24	24 ÷ 1 = 24 ✓
2 × 12	24 ÷ 2 = 12 ✓
3 × 8	24 ÷ 3 = 8 ✓
4 × 6	24 ÷ 4 = 6 ✓

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Multiples of 6: 6, 12, 18, 24, …

Prime: exactly two factors (1 and itself) — 2, 3, 5, 7… 1 is not prime. Composite: more than two factors — 4, 6, 8…

Level 2 — HCF, LCM, and prime factorisation

Prime factorisation: 36 = 2 × 2 × 3 × 3 = 2² × 3².

Idea	Meaning	Example (12 and 18)
HCF	Largest common factor	Factors: {1,2,3,4,6,12} & {1,2,3,6,9,18} → HCF = 6
LCM	Smallest common multiple	Multiples: 12,24,36… & 18,36… → LCM = 36
Co-prime	HCF = 1	8 and 15 share no factor except 1

Rules from prime factors: HCF → lowest power of each common prime; LCM → highest power of each prime that appears. For two numbers a, b: HCF × LCM = a × b.

Worked example

Find HCF and LCM of 24 and 36.

Prime factorisation:
  24 = 2³ × 3
  36 = 2² × 3²

HCF = 2² × 3 = 12   (lowest powers of common primes)
LCM = 2³ × 3² = 72  (highest powers of all primes)

Check: 12 × 72 = 864 = 24 × 36 ✓

Real-life: Two bells ring every 8 s and 12 s. They ring together again after LCM(8, 12) = 24 s.

Common mistakes

Mistake	Why it happens	Fix
Listing 0 as a factor	Confusing multiples with factors	0 cannot divide any number
Swapping HCF and LCM	Similar wording in word problems	"Largest dividing both" → HCF; "smallest divisible by both" → LCM
Forgetting factor 1	Only listing "interesting" factors	1 divides every whole number
Missing factor pairs	Stopping after finding one factor	If 2 divides 30, so does 15

Quick check

List all factors of 48. How many are there?
Find HCF and LCM of 15 and 25 using prime factorisation.
Are 14 and 15 co-prime? Explain briefly.
Two buses return to a depot every 6 min and 9 min. After how many minutes do they meet again?

Open the Practice tab for graded questions on Factors & Multiples.

Factors Multiples