LCM (Lowest Common Multiple)
$$\text{LCM}(a,\,b) = \frac{a \times b}{\gcd(a,\,b)}$$
What is LCM (Lowest Common Multiple)?
The LCM (Lowest Common Multiple) of two positive integers is the smallest positive integer that is a multiple of both. LCM(4, 6) = 12 because 12 is the smallest number appearing in both the multiples of 4 (4, 8, 12, 16...) and the multiples of 6 (6, 12, 18...).
The fastest way to compute LCM uses the GCD identity: LCM(a, b) = (a × b) ÷ GCD(a, b). For LCM(4, 6): GCD(4, 6) = 2, so LCM = (4 × 6) ÷ 2 = 12. When a and b are coprime (GCD = 1), LCM equals their product.
In fraction arithmetic, the LCM of two denominators is the LCD - the smallest number that allows both fractions to be expressed with a shared denominator so their numerators can be added or subtracted.
LCM (Lowest Common Multiple) explained to a beginner
Bus A comes every 4 minutes. Bus B comes every 6 minutes. Both leave together at time zero. When is the next time they leave at the same moment? Multiples of 4: 4, 8, 12... Multiples of 6: 6, 12... First overlap: 12 minutes. That is LCM(4, 6) = 12.
In fractions the idea is identical: LCM of two denominators finds the smallest slot where both denominators fit evenly, so you can rewrite both fractions in same-sized pieces before adding.
When to use LCM (Lowest Common Multiple)
Use LCM when finding the LCD for fraction addition or subtraction, when solving scheduling problems (next coincidence of two repeating events), or when scaling quantities to the smallest common unit. LCM and GCD together are the two key tools in elementary number theory.
Worked examples for LCM (Lowest Common Multiple)
This table quickly gives you the overview you need to understand LCM (Lowest Common Multiple) and its most important comparisons.
| a | b | GCD(a,b) | LCM = a×b÷GCD | Check |
|---|---|---|---|---|
| 4 | 6 | 2 | 12 | Multiples of 4 and 6 first meet at 12 |
| 3 | 5 | 1 | 15 | Coprime: LCM = product |
| 6 | 9 | 3 | 18 | Multiples of 6 and 9 first meet at 18 |
| 5 | 10 | 5 | 10 | 10 is already a multiple of 5 |
| 7 | 13 | 1 | 91 | Coprime: LCM = 7×13 |
Common pitfalls
LCM(a, b) × GCD(a, b) = a × b is a useful check: if you know three of the four values you can derive the fourth. Also: LCM(a, a) = a and LCM(1, n) = n. For three or more numbers, compute pairwise: LCM(a, b, c) = LCM(LCM(a, b), c).
Frequently asked questions about LCM (Lowest Common Multiple)
What is the difference between LCM and GCD?
GCD is the largest factor shared by two numbers - used to simplify fractions. LCM is the smallest multiple shared by two numbers - used to find a common denominator. They are linked: GCD(a, b) × LCM(a, b) = a × b.
When does LCM equal the product a × b?
When a and b are coprime - they share no common factor other than 1. Examples: LCM(3, 5) = 15, LCM(7, 11) = 77. When GCD > 1, LCM is smaller than the product.
How does LCM differ from LCD?
LCD (Lowest Common Denominator) is the same concept as LCM applied specifically to the denominators of fractions. LCD(b, d) = LCM(b, d). LCM is the general term; LCD is used in the fraction arithmetic context.