Highest common factor and lowest common multiple are often taught as two procedures. They become much easier to remember when both are seen as questions about shared prime structure.
The HCF keeps only the prime factors two or more numbers definitely share. The LCM keeps enough prime factors to contain every number completely.
Start with prime factorisation
Consider 72 and 120.
72 = 2³ × 3²
120 = 2³ × 3 × 5HCF: take the shared minimum powers
Both numbers contain 2³. Both contain at least one factor 3. Only 120 contains 5.
HCF = 2³ × 3 = 24The HCF is the largest number that divides both exactly.
LCM: take the required maximum powers
To build a number divisible by both 72 and 120, we need the largest required power of every prime appearing in either number.
LCM = 2³ × 3² × 5 = 360360 contains all of 72’s prime structure and all of 120’s prime structure.
Why minimum for HCF and maximum for LCM?
A common factor cannot use more copies of a prime than the least-equipped number contains. That is why HCF uses the smaller exponent.
A common multiple must contain enough copies of every prime to build each original number. That is why LCM uses the larger exponent.
Worked example 1
84 = 2² × 3 × 7
126 = 2 × 3² × 7HCF = 2 × 3 × 7 = 42.
LCM = 2² × 3² × 7 = 252.
Worked example 2: three numbers
24 = 2³ × 3
36 = 2² × 3²
60 = 2² × 3 × 5HCF = 2² × 3 = 12.
LCM = 2³ × 3² × 5 = 360.
Choosing HCF or LCM from context
Use HCF when a quantity must be split into the largest equal groups or pieces with no remainder. Use LCM when repeating cycles must meet again or when finding the smallest quantity divisible by several values.
For example, ribbons 84 cm and 126 cm long are cut into equal longest pieces. The piece length is HCF(84,126) = 42 cm.
Two lights flash every 12 seconds and 18 seconds. They flash together again after LCM(12,18) = 36 seconds.
A useful identity for two numbers
For two positive integers a and b:
HCF(a,b) × LCM(a,b) = a × bThis identity provides a strong check and follows directly from how minimum and maximum prime exponents combine.
Common errors
Taking the larger exponent for HCF. A factor must fit inside every number, so use the minimum exponent.
Ignoring a prime that appears in only one number when finding LCM. The LCM must contain every required prime.
Choosing HCF because the numbers are being “shared”. Read the question’s structure: largest equal size suggests HCF; first common recurrence suggests LCM.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| HCF larger than one input | Meaning lost | Check that a common factor must divide every input |
| LCM not divisible by an input | Prime coverage incomplete | Rebuild with maximum exponents |
| Uses HCF in cycle problem | Context selection | Ask whether the problem seeks a common size or a common recurrence |
Practice
- Find the HCF and LCM of 48 and 180.
- Find the HCF and LCM of 90, 120 and 150.
- Two buses depart every 24 minutes and 36 minutes. If they leave together now, when do they next leave together?
- Two ropes 96 m and 144 m are cut into equal longest pieces. Find the piece length.
Answers
1. HCF 12, LCM 720. 2. HCF 30, LCM 1800. 3. 72 minutes. 4. 48 m.
Connected routes
Use Prime Factors as the Hidden Structure of Whole Numbers for the decomposition foundation and Indices: Why the Laws Work for exponent structure. Return to the Mathematics Learning Hub.