How to Find the Least Common Multiple
In the spring of 2024, parts of the central United States experienced something that hadn't happened since 1803: two broods of periodical cicadas emerged at the same time. One brood appears every 13 years; the other every 17 years.
When do 13-year and 17-year cycles line up? After the least common multiple of 13 and 17, which is 221 years.
The LCM answers every "when do these repeating things coincide?" question, from cicadas to bus schedules to adding fractions. Here's how to find it.
Prime Cycles Almost Never Meet
It's natural to think cycles of similar lengths would line up often. The opposite can be true. Cycles of 12 and 18 years would coincide every 36 years, because they share factors. Cycles of 13 and 17 years share no factors at all, so they coincide only every 221 years.
Biologists have suggested that prime-numbered life cycles may help periodical cicadas avoid predators whose population cycles are shorter. A predator with a 2-, 3-, 4- or 6-year cycle rarely lines up with a 13- or 17-year emergence.
What Is the LCM?
The least common multiple (LCM) of two or more numbers is the smallest positive number that all of them divide into evenly.
For 4 and 6:
Multiples of 4: 4, 8, 12, 16, 20, 24, …
Multiples of 6: 6, 12, 18, 24, 30, …
Common multiples: 12, 24, 36, … The smallest is 12.
Method 1: List Multiples
Best for small numbers.
Example: LCM of 8 and 12
8: 8, 16, 24, 32, 40, 48
12: 12, 24, 36, 48
The first match is 24. The multiplication tables make listing quick.
Method 2: Prime Factorization
Best for several numbers or larger numbers.
- Write each number as a product of primes.
- Take every prime that appears in any of the numbers, with its highest exponent.
- Multiply.
Example: LCM of 4, 6 and 15
4 = 2²
6 = 2 × 3
15 = 3 × 5
Highest powers: 2², 3¹, 5¹.
LCM = 2² × 3 × 5 = 60
Compare with the GCF, which uses the lowest powers of only the shared primes. Find factorizations with the prime factorization tool.
Method 3: Use the GCF
Best for two large numbers. For any two positive integers:
LCM(a, b) = (a × b) ÷ GCF(a, b)
Example: LCM of 12 and 18
GCF(12, 18) = 6
LCM = (12 × 18) ÷ 6 = 216 ÷ 6 = 36
Because Euclid's algorithm finds the GCF so quickly, this is how computers calculate LCMs. Check with the LCM calculator and the GCF calculator. See How to Find the Greatest Common Factor.
For three or more numbers, apply it step by step: LCM(4, 6, 15) = LCM(LCM(4, 6), 15) = LCM(12, 15) = 60.
Everyday Uses of the LCM
Adding Fractions
To add 5/12 + 7/18, use the least common denominator, which is the LCM of 12 and 18:
5/12 = 15/36
7/18 = 14/36
Sum = 29/36
Schedules
One bus comes every 12 minutes and another every 18 minutes. If they both leave at 8:00, they'll next leave together at 8:36.
Gears and Rotations
A 12-tooth gear meshes with an 18-tooth gear. The same pair of teeth meets again after 36 teeth have passed: 3 turns of the small gear and 2 of the large.
Buying in Packs
Hot dogs come in packs of 8, buns in packs of 12. To have exactly the same number of each, buy 24 of each: 3 packs of hot dogs and 2 packs of buns.
See more in Real-Life Uses of LCM and GCF.
An Insider Reference: The 2024 Cicada Double Emergence
In 2024, Brood XIII (a 17-year brood centered in northern Illinois) and Brood XIX (a 13-year brood widespread across the Southeast and Midwest) emerged in the same year. Entomologists noted the last time these two particular broods had emerged together was 1803, and the next time will be around 2245.
Researchers have long debated why periodical cicadas use prime cycles. One explanation, discussed by biologists and mathematicians alike, is that prime periods minimize overlap with predator cycles and with other cicada broods, reducing competition and hybridization. Whatever the biology, the timing follows straight from the LCM: LCM(13, 17) = 221.
Two Concepts Worth Knowing
Multiple
A multiple of a number is that number times a whole number. The multiples of 7 are 7, 14, 21, … Every number has infinitely many multiples.
Least Common Denominator
The least common denominator of several fractions is the LCM of their denominators. It's the smallest denominator that lets you add or compare the fractions directly.
Quick Answer: How Do You Find the Least Common Multiple?
For small numbers, list multiples of each and find the smallest one they share. Using prime factorization, multiply every prime that appears, each raised to its highest power. For two numbers, you can also use LCM(a, b) = a × b ÷ GCF(a, b). For example, LCM(12, 18) = 216 ÷ 6 = 36.
Try Them Yourself
- LCM Calculator: find the LCM of any numbers
- GCF Calculator: the partner for Method 3
- Prime Factorization Tool: factors for Method 2
- Multiplication Tables: list multiples quickly
- Is 17 Prime?: the cicadas' prime cycle
- Number Theory Formulas: the GCF–LCM identity
Pick two events in your life that repeat on different schedules, like a monthly bill and a biweekly paycheck. Use the LCM to find when they'll next land on the same day.