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.

Beyond fractions, the LCM shows up anywhere repeating cycles need to be synchronised as efficiently as possible. Operating systems use it to schedule recurring background tasks without wasting cycles; sound engineers use it to work out the shortest loop length that lets two rhythms of different lengths repeat together cleanly; and manufacturing lines use it to time conveyor belts moving at different speeds so that parts arrive together at exactly the right moment, as often as possible without waste.

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: Find the LCM

Find the LCM