LCM Calculator

What the LCM is

Multiples of 4 are 4, 8, 12, 16, 20, 24 …; multiples of 6 are 6, 12, 18, 24 …. The first number on both lists, 12, is the least common multiple. Listing multiples works for small numbers — the calculator shows the lists when the LCM is no more than 12 times the smallest number — but for anything bigger, prime factors are faster and never miss.

Three ways to find it

1. Prime factorization. Factor each number, then take each prime at its highest power:

The 2² comes from 12, the 3 from any of them, and the 5² from 75. The result table shows which number each power was taken from.

2. The GCF shortcut. For two numbers, LCM = a × b ÷ GCF. The greatest common factor of 24 and 36 is 12, so the LCM is 864 ÷ 12 = 72. For more numbers, apply it in pairs: LCM(a, b, c) = LCM(LCM(a, b), c). The calculator uses this pairwise method with exact whole-number arithmetic, so even the LCM of 1 through 20, 232,792,560, comes out exactly.

3. Listing multiples. Write out multiples of the largest number and stop at the first one the others divide: 75, 150, 225, 300 — 300 ÷ 12 = 25 and 300 ÷ 15 = 20, so 300 it is.

Least common denominator for fractions

To add or compare fractions, rewrite them over a common denominator; the smallest one is the LCM of the denominators. For 1/4 + 5/6 + 7/15 the LCD is 60:

Now add the tops: 93/60, which simplifies to 1 11/20. Switch the calculator to LCD of fractions to get the rewritten fractions at once, or add, subtract and simplify them in one step with the fraction calculator.

When the LCM comes up

• Schedules: two buses leave together, one every 12 minutes and one every 15; they leave together again after LCM(12, 15) = 60 minutes. To add the times to a clock, use the time calculator.
• Packs that come in different sizes: hot dogs in 10s and buns in 8s match up at 40 of each — 4 packs of hot dogs and 5 of buns.
• Gears and cycles: gears with 12 and 18 teeth line up the same two teeth every 36 teeth, after 3 turns of the small gear and 2 of the large one.

LCM and GCF

The greatest common factor is the largest number that divides into every number in the list — the opposite question. For 12, 15 and 75 the GCF is 3. Numbers with a GCF of 1, such as 8, 9 and 35, have an LCM equal to their product. The result shows the GCF beside the LCM. For powers and long products, the scientific calculator helps check the arithmetic.