Topic A1

Multiples & Factors

Understand what factors and multiples are, then learn how to find HCF and LCM using the division ladder method.

A. Key Definitions

Factor
A number that divides another number exactly — with no remainder left over.
Example Factors of 12 are: 1, 2, 3, 4, 6, 12
12÷1=12 ✓   12÷2=6 ✓   12÷3=4 ✓   12÷4=3 ✓   12÷5=2.4 ✗   12÷6=2 ✓
Multiple
What you get when you multiply a number by 1, 2, 3, 4 … Multiples never end.
Example Multiples of 5 are: 5, 10, 15, 20, 25, 30 …
5×1=5   5×2=10   5×3=15   and so on.
HCF — Highest Common Factor
The largest number that divides into two numbers exactly. Found by breaking each number down using prime division, then picking shared primes with the smaller power.
Example HCF of 12 and 18 = 6
6 is the biggest number that divides both 12 and 18 with no remainder.
LCM — Least Common Multiple
The smallest number that both numbers divide into exactly. Found by dividing both numbers together through primes until you reach 1, then multiplying all the divisors.
Example LCM of 4 and 6 = 12
12 is the smallest number that both 4 and 6 go into exactly.
Quick rule: Factors are always ≤ the number. Multiples are always ≥ the number.

B. Finding HCF — Division Ladder

For HCF, divide each number separately by prime numbers until you reach 1. Then pick the prime factors that appear in both — using the smaller power — and multiply them.

Example 1 — HCF of 12 and 18
1Divide each number separately by primes:
Break down 12
212
26
33
1
12 = 2² × 3
Break down 18
218
39
33
1
18 = 2 × 3²
2Primes in both: 2 and 3
3Smaller power of 2: 2¹  |  Smaller power of 3: 3¹
4HCF = 2 × 3 = 6
HCF(12, 18) = 6  — check: 12÷6=2 ✓  18÷6=3 ✓
Example 2 — HCF of 24 and 36
Break down 24
224
212
26
33
1
24 = 2³ × 3
Break down 36
236
218
39
33
1
36 = 2² × 3²
2Primes in both: 2 and 3. Smaller powers: 2² and 3¹
3HCF = 4 × 3 = 12
HCF(24, 36) = 12
⚠ Watch out: For HCF, after you find the shared primes, always use the smaller power — not the bigger one.

C. Finding LCM — Division Ladder

For LCM, divide both numbers together in one ladder. Keep dividing by a prime as long as it divides at least one number. When a number can't be divided, bring it down as-is. Stop when both reach 1. Then multiply all the left-side divisors.

Example 1 — LCM of 12 and 16
Divide both together
212 , 16
26 , 8
23 , 4
23 , 2
33 , 1
1 , 1
LCM = 2×2×2×2×3 = 48
← divide both by 2
← divide both by 2
← divide both by 2
← only 2 divides by 2; bring 3 down
← only 3 divides by 3; bring 1 down
← both reached 1, stop
✓Multiply all left-side numbers: 2×2×2×2×3 = 48
LCM(12, 16) = 48  — check: 48÷12=4 ✓  48÷16=3 ✓
Example 2 — LCM of 4 and 6
Divide both together
24 , 6
22 , 3
31 , 3
1 , 1
LCM = 2×2×3 = 12
← both divide by 2
← only 2 divides by 2; bring 3 down
← only 3 divides by 3; bring 1 down
← both 1, stop
LCM(4, 6) = 12
HCF — remember
Separate ladders
Shared primes, smaller power
Always ≤ both numbers
LCM — remember
One shared ladder
Multiply all left-side numbers
Always ≥ both numbers

D. The Golden Relationship

Golden Rule HCF × LCM = First number × Second number Missing number = (HCF × LCM) ÷ known number
Example 1 — Verify with 4 and 6
1HCF(4,6)=2  |  LCM(4,6)=12
2HCF × LCM = 2 × 12 = 24
34 × 6 = 24 ✓ Same!
Rule confirmed
Example 2 — HCF=4, LCM=48, one number=12. Find the other.
1Missing = (HCF × LCM) ÷ known
2= (4 × 48) ÷ 12 = 192 ÷ 12 = 16
The other number is 16
⚡ MCQ shortcut: missing number = (HCF × LCM) ÷ known number

Quick MCQ Revision

RuleRemember
HCFSeparate ladders → shared primes × smaller power
LCMOne shared ladder → multiply all left-side numbers
HCF × LCM= product of the two numbers
Missing number(HCF × LCM) ÷ known number
HCF is always≤ both numbers
LCM is always≥ both numbers
NextPrime & Composite Numbers
Key