GCF vs LCM: Work Through 18 and 24

A shared divisor and a shared multiple

The greatest common factor, also called the greatest common divisor, is the largest positive integer dividing both inputs. The least common multiple is the smallest positive integer that is a multiple of both. For 18 and 24, the GCF is 6 and the LCM is 72.

The factor lists make the first result visible. Eighteen has factors 1, 2, 3, 6, 9 and 18. Twenty-four has factors 1, 2, 3, 4, 6, 8, 12 and 24. Their shared factors are 1, 2, 3 and 6.

Use minimum and maximum exponents

Prime factorization gives 18 = 2 × 3² and 24 = 2³ × 3. For the GCF, take each shared prime with the smaller exponent: 2 × 3 = 6. For the LCM, take every prime with the larger exponent: 2³ × 3² = 72.

The product check is 6 × 72 = 432 and 18 × 24 = 432. For two positive integers, GCF × LCM equals their product. If your answers fail this check, inspect the exponents or multiplication.

Choose the operation from the problem

To divide 18 red items and 24 blue items into the greatest number of identical groups with none left over, use the GCF: six groups. To find when cycles of 18 and 24 equal time units next coincide after a common start, use the LCM: 72 units.

The numerical inputs can be identical while the question changes. Write what the result is supposed to represent before selecting a formula.

Check the mathematical definitions

For fractions, percentages and worked arithmetic, see OpenStax Prealgebra. For number definitions, see OpenStax on prime and composite numbers and Wolfram MathWorld on digital roots. Worked examples on this page are calculated for this guide.