HCF & LCM
AdvancedHCF is for sharing (dividing things up, simplifying); LCM is for things happening together (buses meeting, common denominators). Don't mix them up.
After this lesson, you can:
- Factor: divides evenly; Multiple: multiplied by integer
- HCF: highest common factor
- LCM: lowest common multiple
Before this lesson you should know: Place value & the four operations
1Summary 重點摘要
HCF is for sharing (dividing things up, simplifying); LCM is for things happening together (buses meeting, common denominators). Don't mix them up.
2Vocabulary cards Vocabulary
F
Factor因數
Number that divides another evenly.
M
Multiple倍數
Product of a number × integer.
H
Highest common factor (HCF)最大公因數
Largest shared factor.
L
Lowest common multiple (LCM)最小公倍數
Smallest shared multiple.
P
Prime number質數
Number with only 1 and itself as factors.
P
Prime factorization質因數分解
Break into prime factors.
C
Common factor公因數
Shared factor.
C
Common multiple公倍數
Shared multiple.
3Step by step Steps
- 1. What is a factor?Example: What are the factors of 12?
12 ÷ 1 = 12 ✓
12 ÷ 2 = 6 ✓
12 ÷ 3 = 4 ✓
12 ÷ 4 = 3 ✓
12 ÷ 6 = 2 ✓
12 ÷ 12 = 1 ✓
So factors of 12: 1, 2, 3, 4, 6, 12 (6 factors). - 2. What is a multiple?Example: What are the multiples of 3?
3 × 1 = 3
3 × 2 = 6
3 × 3 = 9
3 × 4 = 12
3 × 5 = 15
So multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24... (infinite). - 3. Highest common factor (HCF)Example: HCF of 12 and 18?
Step 1: List factors
12: 1, 2, 3, 4, 6, 12
18: 1, 2, 3, 6, 9, 18
Step 2: Find common
Common: 1, 2, 3, 6
Step 3: Find highest
Highest is 6.
So HCF(12, 18) = 6. - 4. Lowest common multiple (LCM)Example: LCM of 4 and 6?
Step 1: List multiples
4: 4, 8, 12, 16, 20, 24...
6: 6, 12, 18, 24, 30...
Step 2: Find common
Common: 12, 24, 36...
Step 3: Find lowest
Lowest is 12.
So LCM(4, 6) = 12. - 5. Prime factorizationExample: Factorize 12.
12 ÷ 2 = 6
6 ÷ 2 = 3
3 ÷ 3 = 1
So 12 = 2 × 2 × 3.
Prime factors: 2, 2, 3. - 6. HCF by prime factorizationExample: HCF of 12 and 18?
Step 1: Prime factors
12 = 2 × 2 × 3
18 = 2 × 3 × 3
Step 2: Find common prime factors
Common: 2, 3
Step 3: Multiply
2 × 3 = 6.
So HCF(12, 18) = 6. - 7. LCM by prime factorizationExample: LCM of 4 and 6?
Step 1: Prime factors
4 = 2 × 2
6 = 2 × 3
Step 2: Take all primes (repeat the one with more)
2 (twice), 3 (once)
Step 3: Multiply
2 × 2 × 3 = 12.
So LCM(4, 6) = 12.
4Diagrams Diagrams
5Worked example Example
Example: find the HCF and LCM of 12 and 18.
- Prime factorise: 12 = 2 × 2 × 3, 18 = 2 × 3 × 3
- HCF: shared primes once each → 2 × 3 = 6
- LCM: every prime at its largest count → 2 × 2 × 3 × 3 = 36
- Check HCF: 12 ÷ 6 = 2 and 18 ÷ 6 = 3, both exact ✓
- Formula check: HCF × LCM = 6 × 36 = 216 and 12 × 18 = 216 ✓
- Answer: HCF = 6, LCM = 36 — HCF uses shared primes, LCM uses all primes at highest powers.
6Common mistakes Mistakes
✗ Confusing factor and multiple.
✓ Factor: divides; Multiple: multiplied.
✗ Confusing HCF and LCM.
✓ HCF is highest factor; LCM is lowest multiple.
✗ Prime factorization incomplete.
✓ Keep dividing until all are primes.
✗ 1 is a prime?
✓ 1 is NOT prime (prime must be > 1).
7Quick check Quick check
Q1How many factors does 12 have?
Answer: 6 (1, 2, 3, 4, 6, 12).
Did you get it right?
Q2HCF(12, 18) = ?
Answer: 6.
Did you get it right?
Q3LCM(4, 6) = ?
Answer: 12.
Did you get it right?
Q4Prime factorization of 12?
Answer: 2 × 2 × 3.
Did you get it right?
Q5Is 2 a prime number?
Answer: Yes (only even prime).
Did you get it right?
8Recap Recap
- Factor: divides evenly; Multiple: multiplied by integer
- HCF: highest common factor
- LCM: lowest common multiple
- Prime factorization: break into primes
- Prime factorization is fastest way to find HCF/LCM