These two sound like school-only ideas, but you use them whenever you simplify a fraction, tile a floor with equal squares, or work out when two repeating events coincide again.
Greatest common factor (GCF)
The GCF of two numbers is the largest whole number that divides both. For 12 and 18, the factors of 12 are 1, 2, 3, 4, 6, 12 and the factors of 18 are 1, 2, 3, 6, 9, 18. The largest shared one is 6.
Least common multiple (LCM)
The LCM is the smallest number that both divide into. Multiples of 12 run 12, 24, 36, 48 and multiples of 18 run 18, 36, 54, so the first match is 36.
How they connect
For two numbers, GCF × LCM = a × b. Check it: 6 × 36 = 216, and 12 × 18 = 216. This gives a fast route to the LCM once you know the GCF: multiply the numbers and divide by the GCF.
For bigger numbers, Euclid’s method finds the GCF without listing factors. Divide the larger by the smaller, then replace the larger with the remainder and repeat until the remainder is zero. For 48 and 180: 180 = 3 × 48 + 36; 48 = 1 × 36 + 12; 36 = 3 × 12 + 0. The GCF is 12, and the LCM is 48 × 180 ÷ 12 = 720.
Using the calculator
- Enter two whole numbers in the GCF and LCM Calculator.
- Read both results on one line, for example “GCF(12, 18) = 6 — LCM(12, 18) = 36”.
It handles two non-zero whole numbers at a time. For three or more numbers, find the GCF or LCM of two, then combine that result with the next number.
Where this comes up
- Simplifying fractions: divide top and bottom by their GCF.
- Adding fractions: the LCM of the denominators is the smallest common denominator.
- Scheduling: two buses running every 12 and 18 minutes leave together every 36 minutes.
Frequently asked questions
What if the GCF is 1?
Then the numbers are coprime, and their LCM is simply their product.
Does GCF mean the same as HCF?
Yes. Highest common factor (HCF) and greatest common divisor (GCD) are other names for the same thing.