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ACT® · Math: Number and Quantity

Greatest Common Factor and Least Common Multiple

The greatest common factor (GCF) is the largest number that divides both numbers; the least common multiple (LCM) is the smallest number both divide into. Prime factorization finds both at once.

Watch it drawn4 questionsPractice

Break each number into its prime factors and line them up. The GCF takes the lowest power of every prime the two numbers share — it's everything they have in common. The LCM takes the highest power of every prime that shows up in either number — it's enough of everything to cover both. One answers "what divides evenly into both," the other answers "what do both divide evenly into."

This is why gears with 12 and 18 teeth sync up every 36 teeth. 12 = 22 x 3 and 18 = 2 x 32. The GCF is 2 x 3 = 6, the LCM is 22 x 32 = 36. The gears share a rhythm of 6 teeth, and the full cycle where both return to start is 36 teeth — their LCM. The same logic runs any "when do these two repeating things line up again" question, from bus schedules to blinking lights.

There's a shortcut tying GCF and LCM together: for any two positive integers, GCF times LCM equals the product of the two numbers. That's because every prime factor gets counted exactly once between the lowest-power version (GCF) and the highest-power version (LCM), together matching the two original factorizations. This shows up on the number and quantity portion alongside complex numbers, vectors, and matrices — all places where the test checks whether you can manipulate structure, not just compute.

Key terms

Greatest common factor (GCF)
The largest number that divides evenly into two or more given numbers.
Least common multiple (LCM)
The smallest number that two or more given numbers all divide into evenly.
Prime factorization
Writing a number as a product of prime numbers, each raised to a power.
Coprime
Two numbers whose greatest common factor is 1, meaning they share no prime factors.

Watch it explained

  1. How it worksHow do the prime factorizations of two numbers hand you the GCF and the LCM at a glance?
  2. Worked exampleFind the GCF and the LCM of 24 and 60 from their prime factorizations.
  3. Worked exampleFind when two buses leaving at 9 am on 15 minute and 25 minute loops next depart together.
  4. Common mistakeWhy isn't the least common multiple of two numbers simply the two numbers multiplied together?

Check yourself

Your turnmedium1 of 3
Two positive integers have a greatest common factor of 7 and a least common multiple of 7 times 15. What is the product of the two integers?

End of topic.

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