GCF and LCM of 48 and 72: Grouping vs Repeating Schedules

Two answers to two different questions

The greatest common factor (GCF) of 48 and 72 is 24. It is the largest positive integer dividing both without a remainder. The least common multiple (LCM) is 144, the smallest positive value divisible by both.

Use 24 when cutting two lengths into the largest equal pieces or forming identical groups with nothing left over. Use 144 when looking for a common repeating interval. Factors divide the starting numbers; multiples are produced by multiplying them.

Find the GCF from complete factor lists

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

The common factors are 1, 2, 3, 4, 6, 8, 12, 24. The largest is 24. Checking 48 ÷ 24 = 2 and 72 ÷ 24 = 3 confirms that neither division leaves a remainder.

Find the LCM and verify it

For positive integers, GCF × LCM = the product of the two numbers. Therefore LCM = (48 × 72) ÷ 24 = 144. Divide by the GCF before multiplying if you want smaller intermediate values.

Check: 144 ÷ 48 = 3, and 144 ÷ 72 = 2. These are whole numbers. The product check is 24 × 144 = 3456. Simply multiplying the inputs gives a common multiple, but here it is not the least one because they share a factor greater than 1.

Apply each result to a different situation

Two ribbons measure 48 cm and 72 cm. The longest equal pieces that use both ribbons completely are 24 cm each. That produces 2 pieces from the first ribbon and 3 from the second, or 5 pieces in all.

Two practice reminders begin together and recur every 48 and 72 minutes. They next coincide after 144 minutes, assuming exact intervals and no interruptions. If they start at different times, you need their starting offsets as well; the LCM alone does not settle that problem.

Try a scaled example

Double both starting numbers to 96 and 144. Their GCF becomes 48, and their LCM becomes 288. The number of ribbon pieces in the grouping example remains the same because all lengths were scaled equally.

Before calculating, name the unit and decide whether you are seeking a divisor or a shared multiple. That choice is usually more important than the arithmetic itself.

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.