Factors and Multiples Masterclass

1. Definitions

Factor: A number that divides exactly into another number (no remainder). Factors are usually smaller than the number.

Multiple: The result of multiplying a number by an integer (times tables). Multiples are usually larger than the number.

Example using the number 12:

2. HCF: Highest Common Factor

The largest number that divides exactly into both numbers.

Method A: Listing (Small Numbers)

Find HCF of 12 and 18
Factors of 12: \(1, 2, 3, 4, \mathbf{6}, 12\)
Factors of 18: \(1, 2, 3, \mathbf{6}, 9, 18\)
The biggest number in both lists is 6.

Method B: Prime Factors (Large Numbers)

Find HCF of 120 and 168
Step 1: Write as product of primes \(120 = 2 \times 2 \times 2 \times 3 \times 5 = 2^3 \times 3 \times 5\)
\(168 = 2 \times 2 \times 2 \times 3 \times 7 = 2^3 \times 3 \times 7\)

Step 2: Multiply the Common Primes Both have three 2s and one 3.
$$ \text{HCF} = 2 \times 2 \times 2 \times 3 = 24 $$

3. LCM: Lowest Common Multiple

The smallest number that is in the times tables of both numbers.

Method A: Listing (Small Numbers)

Find LCM of 6 and 8
Multiples of 6: \(6, 12, 18, \mathbf{24}, 30, 36...\)
Multiples of 8: \(8, 16, \mathbf{24}, 32, 40...\)
The first number to appear in both lists is 24.

Method B: Prime Factors (Large Numbers)

Find LCM of 60 and 72
\(60 = 2^2 \times 3 \times 5\)
\(72 = 2^3 \times 3^2\)

The Rule: Take the highest power of every prime number present in either list.
- Highest \(2\) is \(2^3\)
- Highest \(3\) is \(3^2\)
- Highest \(5\) is \(5^1\)

$$ \text{LCM} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360 $$

4. Summary Table

Concept Think... Result Size
HCF "What fits into both?" Smaller than (or equal to) numbers given.
LCM "Where do their times tables meet?" Larger than (or equal to) numbers given.