The Quick Answer
GCD (Greatest Common Divisor): The largest number that divides both numbers evenly.
LCM (Least Common Multiple): The smallest number that both numbers divide into evenly.
| Numbers | GCD | LCM |
|---|---|---|
| 12 and 18 | 6 | 36 |
| 15 and 20 | 5 | 60 |
| 7 and 11 | 1 | 77 |
Finding GCD: The Euclidean Algorithm
The fastest method for finding GCD:
- Divide the larger number by the smaller
- Take the remainder
- Replace the larger with the smaller, smaller with remainder
- Repeat until remainder is 0
- The last non-zero value is the GCD
Example: GCD of 48 and 18
48 ÷ 18 = 2 remainder 12
18 ÷ 12 = 1 remainder 6
12 ÷ 6 = 2 remainder 0
GCD = 6
Example: GCD of 56 and 42
56 ÷ 42 = 1 remainder 14
42 ÷ 14 = 3 remainder 0
GCD = 14
Finding GCD: Prime Factorization
Another method using prime factors:
- Find prime factors of each number
- Identify common factors
- Multiply the common factors together
Example: GCD of 36 and 48
36 = 2 × 2 × 3 × 3 = 2² × 3²
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3
Common: 2² × 3 = 4 × 3 = 12
GCD = 12
Finding LCM
Method 1: Using GCD
The fastest way once you have GCD:
LCM(a, b) = (a × b) / GCD(a, b)
Example: LCM of 12 and 18
GCD(12, 18) = 6
LCM = (12 × 18) / 6 = 216 / 6 = 36
Method 2: Prime Factorization
- Find prime factors of each number
- Take the highest power of each prime
- Multiply them together
Example: LCM of 12 and 18
12 = 2² × 3
18 = 2 × 3²
Take highest powers:
- 2² (from 12)
- 3² (from 18)
LCM = 2² × 3² = 4 × 9 = 36
GCD and LCM Relationship
For any two numbers a and b:
GCD(a, b) × LCM(a, b) = a × b
This is useful for checking your work:
- 12 × 18 = 216
- 6 × 36 = 216 ✓
When GCD = 1
Two numbers with GCD = 1 are called coprime or relatively prime.
Examples:
- 7 and 11 (both prime)
- 8 and 15 (no common factors)
- 14 and 25 (no common factors)
For coprime numbers: LCM(a, b) = a × b
Practical Applications
GCD Uses
| Application | Example |
|---|---|
| Simplifying fractions | 12/18 → 2/3 (divide by GCD 6) |
| Cutting equal pieces | Cut 48cm and 18cm ropes into equal lengths |
| Tiling | Fit tiles evenly in a room |
LCM Uses
| Application | Example |
|---|---|
| Adding fractions | 1/4 + 1/6 needs denominator 12 |
| Scheduling | Events every 3 and 4 days meet at day 12 |
| Gear ratios | When do two gears align? |
Examples
Simplifying Fractions
Simplify 24/36:
GCD(24, 36) = 12
24/36 = (24÷12)/(36÷12) = 2/3
Finding Common Denominators
Add 1/4 + 1/6:
LCM(4, 6) = 12
1/4 = 3/12
1/6 = 2/12
3/12 + 2/12 = 5/12
Multiple Numbers
For three or more numbers, apply iteratively:
GCD(12, 18, 24)
= GCD(GCD(12, 18), 24)
= GCD(6, 24)
= 6
LCM(4, 6, 8)
= LCM(LCM(4, 6), 8)
= LCM(12, 8)
= 24
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