Least Common Multiple (LCM)
The Least Common Multiple (LCM) is the smallest positive number that is a multiple of two or more given numbers. It is the gateway to adding and subtracting fractions with different denominators.
Definition
LCM(a, b) = the smallest positive integer divisible by both a and b.
Finding LCM by Listing
LCM(4, 6)
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24...
Smallest shared: 12. LCM(4,6) = 12.
LCM(8, 12, 15)
8: 8,16,24,32,40,48,56,64,72,80,88,96,104,112,120...
12: 12,24,36,48,60,72,84,96,108,120...
15: 15,30,45,60,75,90,105,120...
LCM = 120
Using LCM to Add Fractions
1/4 + 1/6
LCM(4,6) = 12. Convert: 3/12 + 2/12 = 5/12
LCM Properties
| Property | Example |
|---|---|
| LCM(a,b) = LCM(b,a) | LCM(4,6) = LCM(6,4) = 12 |
| LCM(a,a) = a | LCM(7,7) = 7 |
| If a divides b, LCM(a,b) = b | LCM(4,8) = 8 |
| LCM(a,b) × GCF(a,b) = a × b | 12 × 2 = 4 × 6 = 24 ✓ |
Key Takeaways
- LCM is the smallest number that both (or all) numbers divide into.
- All other common multiples are multiples of the LCM.
- LCM is essential for finding common denominators.
- GCF × LCM = product of the two numbers.
Practice Questions
- Find LCM(9, 12).
- Find LCM(5, 7). (Hint: what does this tell you about 5 and 7?)
- Add 1/6 + 1/9 using the LCM.
- Find LCM(4, 6, 10).
- Two traffic lights change every 45 s and 60 s. They both change at 12:00. When do they next change together?