What is the least common multiple?
The least common multiple is the smallest positive number that every number in your list divides into exactly. For 6 and 8 it’s 24: it’s in the 6 times table and the 8 times table, and nothing smaller is in both.
In the UK it’s often called the lowest common multiple, which means exactly the same thing. When the numbers are the denominators of fractions, the LCM is called the lowest common denominator.
Four ways to find the LCM
List the multiples
Write out each times table until a number turns up in every list. For 6 and 8: 6, 12, 18, 24 and 8, 16, 24. The first shared number, 24, is the LCM. It’s the easiest method to follow, but slow when the LCM is large.
Use prime factors
Split each number into primes, then multiply the highest power of every prime that appears anywhere. For 20 = 2² × 5, 24 = 2³ × 3 and 45 = 3² × 5, take 2³, 3² and 5: 8 × 9 × 5 = 360. Compare this with the GCF, which takes the lowest power of the primes they all share.
Use the ladder (cake) method
Place the numbers in one row. At each stage, choose a prime that divides at least two entries, divide those entries and copy the others into the next row. For 20, 24 and 45, successive divisions by 2, 2, 3 and 5 finish with 1, 2 and 3. Multiplying the side divisors and final row gives 2 × 2 × 3 × 5 × 1 × 2 × 3 = 360. The same layout may be called a division grid, factor box, ladder or cake.
Use the GCF
For two positive numbers, LCM(a, b) = a × b ÷ GCF(a, b). For 28 and 42, the GCF is 14, so the LCM is 1,176 ÷ 14 = 84. With a longer list, combine the first pair, then combine that result with each remaining number. This approach is efficient for large integers because the Euclidean algorithm can find the GCF without first factoring both inputs.
How the LCM and GCF fit together
For any two positive numbers, GCF × LCM = the product of the numbers. For 12 and 18, the GCF is 6 and the LCM is 36, and 6 × 36 = 216 = 12 × 18. That’s why the GCF formula works. The calculator shows this check whenever you enter two numbers, and links to the full GCF working.
Zero, negative numbers and decimals
The only multiple of 0 is 0, so if one of the numbers is 0, the LCM is 0 by convention. School questions normally use positive numbers only. Negative signs are ignored, since the LCM is always given as a positive number: LCM(−4, 6) = 12.
For a finite set of terminating decimals, first multiply every value by the same power of 10 so that all become integers. Find the integer LCM, then reverse the scaling. For 0.4 and 0.6, multiplying both by 10 gives 4 and 6; their LCM is 12, so the scaled-back result is 1.2. The calculator itself accepts whole-number inputs.
Where the LCM is useful
Adding fractions
To add 1/4 and 5/6, rewrite both over the LCM of 4 and 6, which is 12: 3/12 + 10/12 = 13/12. Using the LCM keeps the numbers as small as possible. The fraction calculator links here whenever it finds a common denominator.
When repeating events line up
Suppose two maintenance checks repeat every 14 and 20 days. Starting together today, both next fall on the same day after LCM(14, 20) = 140 days. The same reasoning works for transport intervals, rotating equipment and recurring schedules.
Buying packs that match
Picture frames come in boxes of 6 while matching wall hooks come in packets of 8. The smallest equal quantity is LCM(6, 8) = 24, requiring 4 boxes of frames and 3 packets of hooks.