Primes, Factors, Multiples; HCF and LCM

Key Vocabulary

• Factor of $n$: a whole number that divides $n$ exactly (no remainder). Example: factors of 12 are 1, 2, 3, 4, 6, 12.
• Multiple of $n$: a number in the times-table of $n$. Example: multiples of 4 are 4, 8, 12, 16, …
• Prime number: a number greater than 1 with exactly two factors — 1 and itself. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note: 1 is not prime.

Prime Factor Decomposition (Product of Prime Factors)

Every integer greater than 1 can be written as a product of prime factors. Use a factor tree or repeated division to find them.

Example: Write 360 as a product of prime factors.

$$360 \div 2 = 180,\quad 180 \div 2 = 90,\quad 90 \div 2 = 45,\quad 45 \div 3 = 15,\quad 15 \div 3 = 5$$

$$360 = 2^3 \times 3^2 \times 5$$

Write in index form and check: $8 \times 9 \times 5 = 360$. ✓

Highest Common Factor (HCF)

The HCF of two (or more) numbers is the largest factor shared by all of them.

Method (product of prime factors):

1. Write each number as a product of prime factors.
2. Identify common prime factors.
3. Multiply together the lowest power of each common prime.

Example: Find HCF(72, 180). $$72 = 2^3 \times 3^2$$ $$180 = 2^2 \times 3^2 \times 5$$ Common primes: 2 and 3. Lowest powers: $2^2$ and $3^2$. $$\text{HCF} = 2^2 \times 3^2 = 4 \times 9 = 36$$

Lowest Common Multiple (LCM)

The LCM of two (or more) numbers is the smallest positive multiple shared by all of them.

Method (product of prime factors):

1. Write each number as a product of prime factors.
2. Identify all prime factors (from either number).
3. Multiply together the highest power of each prime.

Example: Find LCM(72, 180). $$72 = 2^3 \times 3^2,\quad 180 = 2^2 \times 3^2 \times 5$$ All primes: 2, 3, 5. Highest powers: $2^3$, $3^2$, $5^1$. $$\text{LCM} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360$$

Quick check: $\text{HCF} \times \text{LCM} = 36 \times 360 = 12,960 = 72 \times 180$. ✓

Useful Applications

• HCF is useful when simplifying fractions or sharing things into equal groups.
• LCM is useful for adding fractions (common denominator) or finding when events coincide (e.g. bus timetables).

WJEC Exam Tips

• Always use prime factor trees or repeated division — show all working.
• Circle common factors when comparing prime factorisations.
• Read carefully: HCF asks for the greatest common factor; LCM asks for the smallest common multiple.
• For WJEC Foundation/Intermediate: HCF and LCM typically involve two 2-digit numbers.
• For Higher: may involve three numbers or algebraic contexts.