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Factors, multiples, GCF and LCM

Find the greatest common factor and least common multiple, use them in sharing, scheduling and fraction problems, and factor out a common number. Includes six original questions with worked answers.

By FinishHigh · Published 2026-11-02 · Updated 2026-11-02

Factors divide in; multiples count up

A factor of 12 divides 12 with nothing left over: 1, 2, 3, 4, 6 and 12. A multiple of 12 is what you get when you count by 12: 12, 24, 36, 48 and so on.

So factors of a number are never larger than it, and multiples are never smaller. On the GED® math test this kind of question is in the first part, where no calculator is allowed, so it helps to list factors in pairs: 1 × 12, 2 × 6, 3 × 4.

Greatest common factor (GCF): the biggest equal share

Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The largest number on both lists is 6, so the GCF is 6.

Use it when you split things into identical groups with nothing left over. With 18 apples and 24 oranges, you can make at most 6 identical bags, each with 3 apples and 4 oranges. The GCF also simplifies a fraction in one step: 18/24 = 3/4 after dividing top and bottom by 6.

Least common multiple (LCM): when things line up again

Multiples of 6: 6, 12, 18, 24. Multiples of 8: 8, 16, 24. The first number on both lists is 24, so the LCM is 24.

Use it when two repeating things meet again. One bus leaves every 6 minutes and another every 8 minutes. If both leave at 9:00, they next leave together at 9:24. The LCM is also the smallest common denominator: 1/6 + 3/8 = 4/24 + 9/24 = 13/24.

A shortcut for bigger numbers: prime factors

Break each number into primes. 60 = 2 × 2 × 3 × 5 and 84 = 2 × 2 × 3 × 7.

For the GCF, take only what they share: 2 × 2 × 3 = 12. For the LCM, take every factor, using shared ones once: 2 × 2 × 3 × 5 × 7 = 420. A check that always works for two numbers: GCF × LCM equals the two numbers multiplied together. Here 12 × 420 = 5,040 and 60 × 84 = 5,040.

Factor out a common number

The GCF lets you rewrite a sum. 12 + 18 has a GCF of 6, so 12 + 18 = 6(2 + 3). Multiply it back out to check: 6 × 2 + 6 × 3 = 12 + 18.

The same move works in algebra: 8x + 12 = 4(2x + 3), because 4 is the largest number that divides both 8 and 12.

Which one does the question want?

If the question is about splitting into equal groups, or the largest possible size, think GCF. If it is about the next time things match, or the smallest amount that works for both, think LCM.

Where this comes from: GED Testing Service’s Assessment Guide for Educators: Mathematical Reasoning lists indicator Q.1.b, using multiples and factors, such as the least common multiple, greatest common factor or the distributive property, to rewrite numeric expressions. Its GED math study booklet lists multiples and factors as a skill. We checked both on November 2, 2026.

These sources set the topic. The explanations, numbers and questions here are original FinishHigh practice with invented situations, not GED® test questions. GED® is a registered trademark of the American Council on Education. FinishHigh is independent and not affiliated with or endorsed by it. This lesson teaches one skill. It is not a full course or a readiness result.

Try this next: 6 questions ↓

ORIGINAL PRACTICE

Your turn.

Try each question before opening the explanation. These are FinishHigh practice examples, not GED® test questions.

What is the greatest common factor of 36 and 48?
Show the answer and why
12. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The largest shared one is 12.
What is the least common multiple of 4 and 10?
Show the answer and why
20. Multiples of 4: 4, 8, 12, 16, 20. Multiples of 10: 10, 20. The first shared one is 20.
Hot dogs come 10 to a pack and buns 8 to a pack. What is the smallest number of each you can buy to have exactly the same number of hot dogs and buns?
Show the answer and why
40 of each: 4 packs of hot dogs and 5 packs of buns. The LCM of 10 and 8 is 40.
A teacher has 30 pens and 45 pencils. She wants identical kits with nothing left over and as many kits as possible. How many kits, and what goes in each?
Show the answer and why
15 kits, each with 2 pens and 3 pencils. The GCF of 30 and 45 is 15.
Rewrite 28 + 42 as a whole number times a sum, using the greatest common factor.
Show the answer and why
14(2 + 3). The GCF of 28 and 42 is 14. Check: 14 × 2 + 14 × 3 = 28 + 42 = 70.
Add 5/12 + 1/8.
Show the answer and why
13/24. The LCM of 12 and 8 is 24, so 5/12 = 10/24 and 1/8 = 3/24. Then 10/24 + 3/24 = 13/24.