What are Factors?
A factor of a number is any whole number that divides exactly into it, leaving no remainder.
For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because:
12 ÷ 1 = 1212 ÷ 2 = 612 ÷ 3 = 412 ÷ 4 = 312 ÷ 6 = 212 ÷ 12 = 1
Every number has at least two factors: 1 and itself.
Highest Common Factor (HCF)
The Highest Common Factor (HCF) of two or more numbers is the largest number that is a factor of all the given numbers.
Example: Find the HCF of 18 and 24.
- List the factors of 18:
1, 2, 3, 6, 9, 18. - List the factors of 24:
1, 2, 3, 4, 6, 8, 12, 24. - Identify the common factors (numbers that appear in both lists):
1, 2, 3, 6. - The highest of these common factors is 6.
Answer: The HCF of 18 and 24 is 6.
Prime Numbers & Prime Factorisation
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, etc. (Note: 1 is not a prime number).
Prime factorisation is the process of writing a number as a product of its prime factors.
Example: Find the prime factorisation of 36.
We can use a factor tree:
36
/ \
2 18 (2 is prime)
/ \
2 9 (2 is prime)
/ \
3 3 (3 is prime)
The prime factors are the numbers at the ends of the branches: 2, 2, 3, 3.
Answer: The prime factorisation of 36 is 2 × 2 × 3 × 3 or 2² × 3².
More Worked Examples
1. Finding All Factors of 48
To find all factors of 48, think of pairs of numbers that multiply to make 48:
1 × 48 = 482 × 24 = 483 × 16 = 484 × 12 = 486 × 8 = 48
Factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
2. Finding HCF of 60 and 72
- Factors of 60:
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. - Factors of 72:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. - Common factors:
1, 2, 3, 4, 6, 12. - Highest Common Factor (HCF) is
12.
Answer: The HCF of 60 and 72 is 12.
Exam-Style Factor Problems
1. Tiling a Floor
A rectangular floor measures 240cm by 300cm. It is to be tiled completely with identical square tiles. What is the largest possible side length of these square tiles?
Answer: 60 cm
Explanation: We need to find the largest number that will divide exactly into both 240 and 300. This is the Highest Common Factor (HCF).
- Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240.
- Factors of 300: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300.
- The common factors include 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
- The HCF is 60. So the largest possible side length of the tiles is 60cm.
2. Prime Factor Product
Write 84 as a product of its prime factors.
Answer: 2 × 2 × 3 × 7 or 2² × 3 × 7
Explanation (Factor Tree Method):
84
/ \
2 42 (2 is prime)
/ \
2 21 (2 is prime)
/ \
3 7 (3 and 7 are prime)
The prime factors at the end of the branches are 2, 2, 3, and 7.
Interactive Factor Tools
1. Factor Finder
Factors: ?
2. HCF Finder
HCF: ?
3. Prime Factorisation
Prime Factorisation: ?
Key Takeaways
- Factors are numbers that divide exactly into another number.
- The Highest Common Factor (HCF) is the largest factor shared by two or more numbers.
- Prime numbers have exactly two factors: 1 and themselves.
- Prime factorisation is writing a number as a product of its prime factors.
- These concepts are useful for simplifying fractions, solving problems involving equal sharing, and more.