Find the least common multiple (LCM) of two or more numbers
The least common multiple (LCM) is the smallest positive integer that is divisible by two or more numbers. For example, the LCM of 4 and 6 is 12 because 12 is the smallest number that both 4 and 6 divide into evenly.
LCM is essential for adding fractions with different denominators, scheduling recurring events, and solving problems involving cycles. When you find a common denominator for 1/4 + 1/6, you're finding the LCM of 4 and 6.
LCM and GCF are inversely related: for any two numbers a and b, LCM(a,b) × GCF(a,b) = a × b. This relationship provides an efficient way to calculate LCM once you know the GCF.
List multiples of each number until you find the first common one.
Use highest power of each prime factor present in any number.
LCM = (a × b) / GCF(a,b). Fast when GCF is known.
Divide numbers by common primes, multiply all divisors.
Reference table for frequently needed LCM calculations:
| Numbers | LCM | GCF | Product |
|---|---|---|---|
| 2, 3 | 6 | 1 | 6 |
| 4, 6 | 12 | 2 | 24 |
| 3, 5 | 15 | 1 | 15 |
| 6, 8 | 24 | 2 | 48 |
| 12, 15 | 60 | 3 | 180 |
| 4, 6, 8 | 24 | 2 | N/A |
Find LCM of denominators to get common denominator. For 1/4 + 1/6: LCM(4,6)=12, so 3/12 + 2/12 = 5/12.
Events repeating every 4 and 6 days coincide every LCM(4,6)=12 days. Useful for maintenance schedules.
Gears with 12 and 18 teeth align every LCM(12,18)=36 tooth movements. Important in machinery.
Flashing lights at 3 and 5 second intervals both flash together every LCM(3,5)=15 seconds.
Find the smallest positive integer that is divisible by both 12 and 18, useful when aligning two gear cycles or finding a common denominator for 1/12 + 1/18.
ResultLCM(12, 18) = 36
Factor each number: 12 = 2^2 x 3 and 18 = 2 x 3^2. Take the highest power of every prime that appears in either factorization, giving 2^2 x 3^2 = 4 x 9 = 36. Cross-check with the GCF identity: GCF(12, 18) = 6, so LCM = (12 x 18) / 6 = 216 / 6 = 36.
LCM is always at least as large as the largest number. If one number divides the other, LCM equals the larger number. Otherwise, LCM is larger than both.
For coprime numbers (GCF=1), LCM equals the product. LCM(3,7) = 21 because 3 and 7 share no factors except 1.
LCM of fractions = LCM of numerators / GCF of denominators. For 2/3 and 4/5: LCM = LCM(2,4)/GCF(3,5) = 4/1 = 4.
Yes! Find LCM of first two, then LCM of that result with the third, and so on. LCM(4,6,8) = LCM(LCM(4,6),8) = LCM(12,8) = 24.
Every prime p appears to power min(e_a, e_b) in the GCF and max(e_a, e_b) in the LCM. Adding these exponents gives e_a + e_b, which is exactly the exponent of p in a x b. Since this holds for every prime, the products are equal. The identity only works for two numbers; it does not extend directly to three or more.
To add 1/4 + 1/6, you need a single shared denominator. The smallest one is LCM(4, 6) = 12. Rewriting gives 3/12 + 2/12 = 5/12. Using LCM keeps the numbers as small as possible compared with multiplying denominators (which would give 24).
LCM aligns repeating cycles. Two gears with 12 and 18 teeth return to their starting positions every LCM(12, 18) = 36 tooth advances. Calendars use LCM to align lunar and solar cycles, and maintenance schedules use it to find dates when multiple recurring tasks coincide.
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