How to Find the GCF and LCM (With Quick Tricks)

The GCF (greatest common factor) is the largest number that divides two numbers evenly; the LCM (least common multiple) is the smallest number both divide into. For 12 and 18, the GCF is 6 and the LCM is 36. A quick check: GCF × LCM equals the product of the two numbers, since 6 × 36 = 12 × 18 = 216.

GCF and LCM are best friends, but most students mix them up. Once you know what each one does, you’ll know which to use without thinking.

Quick definitions

• GCF (Greatest Common Factor): the biggest number that divides into two or more numbers evenly. Used when splitting things into equal groups.
• LCM (Least Common Multiple): the smallest number that two or more numbers both divide into. Used when finding when two cycles meet up.

Method 1, List the factors / multiples

Works great for small numbers.

GCF of 12 and 18:

• Factors of 12: 1, 2, 3, 4, 6, 12.
• Factors of 18: 1, 2, 3, 6, 9, 18.
• Largest shared: 6.

LCM of 4 and 6:

• Multiples of 4: 4, 8, 12, 16, 20..
• Multiples of 6: 6, 12, 18, 24..
• Smallest shared: 12.

Method 2, Prime factorization

For bigger numbers, factor each into primes.

GCF of 24 and 36:

• \(24 = 2^3 \cdot 3\).
• \(36 = 2^2 \cdot 3^2\).
• For GCF, take the lowest power of each shared prime: \(2^2 \cdot 3 = 12\).

LCM of 24 and 36:

• For LCM, take the highest power of each prime: \(2^3 \cdot 3^2 = 72\).

Method 3, The ladder (division) method

Stack the numbers and divide by common primes:

“ 2 | 24 36 2 | 12 18 3 | 6 9 | 2 3 “

• GCF = product of the left column = \(2 \cdot 2 \cdot 3 = 12\).
• LCM = product of the left column plus the bottom row = \(12 \cdot 2 \cdot 3 = 72\).

The magic identity

\[\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b\]

Use it to check your work.

Common mistakes

• Confusing the two, remember: GCF is for Common (smaller); LCM is for Multiple (bigger).
• Forgetting primes that aren’t shared when computing LCM.

FAQ

How do you find the GCF and LCM of two numbers?

For small numbers, list them out. For the GCF, list the factors of each number and pick the largest shared one: for 12 and 18, the factors in common are 1, 2, 3 and 6, so the GCF is 6. For the LCM, list multiples until one matches: multiples of 12 are 12, 24, 36 and multiples of 18 are 18, 36, so the LCM is 36.

How do you find GCF and LCM using prime factorization?

Break each number into primes, then compare. For the GCF, multiply the shared primes using the lowest power of each. For the LCM, multiply every prime that appears using the highest power. With 24 = 2³ × 3 and 36 = 2² × 3², the GCF is 2² × 3 = 12 and the LCM is 2³ × 3² = 72. A factor tree is an easy way to find the primes.

How do you find the GCF and LCM of 3 numbers?

Prime factorization works best. Take 12 = 2² × 3, 18 = 2 × 3² and 30 = 2 × 3 × 5. The GCF uses only primes all three share, at the lowest power: 2 × 3 = 6. The LCM uses every prime at its highest power: 2² × 3² × 5 = 180. Check: 180 ÷ 12 = 15, 180 ÷ 18 = 10 and 180 ÷ 30 = 6.

How do you find GCF and LCM using continuous division?

Write the numbers side by side and keep dividing both by a prime that goes into each, writing the quotients underneath. Stop when no common prime is left. For 30 and 45, divide by 3 to get 10 and 15, then by 5 to get 2 and 3. The GCF is the product of the divisors, 3 × 5 = 15. The LCM is 15 × 2 × 3 = 90.

How do you find GCF and LCM in word problems?

Look at what the question asks. If you are splitting things into the largest equal groups, use the GCF: 24 apples and 36 oranges can fill at most 12 identical bags, each with 2 apples and 3 oranges. If you need when two repeating events line up again, use the LCM: buses every 6 and 8 minutes leave together again after 24 minutes.

How do you find the GCF and LCM fast?

Find the GCF first, then use the shortcut LCM = (a × b) ÷ GCF. For 8 and 12, the GCF is 4, so the LCM is 96 ÷ 4 = 24. You can also check any answer with GCF × LCM = a × b. Another quick clue: the GCF can never be larger than the smaller number, and the LCM can never be smaller than the larger number.

Word-problem cues for GCF vs. LCM

These keywords almost always tell you which one to use:

GCF cues (you’re dividing into equal groups):

• “What’s the largest..”
• “How many groups can be formed..”
• “Equal piles, no leftover..”
• “Same number of each in each bag..”

LCM cues (you’re finding when two cycles align):

• “When will they meet again..”
• “At what time..”
• “Next time both..”
• “Common denominator..”

Worked example, GCF word problem

A teacher has 36 pencils and 60 erasers. She wants to put them into identical gift bags with no leftovers. What’s the largest number of bags she can make?

This is GCF, “largest number of equal groups.” Factor: \(36 = 2^2 \cdot 3^2\), \(60 = 2^2 \cdot 3 \cdot 5\). GCF = \(2^2 \cdot 3 = 12\). 12 bags, each with 3 pencils and 5 erasers.

Worked example, LCM word problem

Bus A arrives every 12 minutes. Bus B arrives every 18 minutes. If they both arrive at 8:00 AM, what’s the next time they’ll arrive together?

LCM(12, 18). Factor: \(12 = 2^2 \cdot 3\), \(18 = 2 \cdot 3^2\). LCM = \(2^2 \cdot 3^2 = 36\). They’ll both arrive together again at 8:36 AM.

A quick mental check on every answer

• GCF must divide both original numbers.
• LCM must be divisible by both original numbers.
• Always: \(\text{GCF} \le \min(a, b)\) and \(\text{LCM} \ge \max(a, b)\).

If your answer violates either rule, you’ve made a mistake somewhere.

The relationship between GCF and LCM

Here’s a beautiful identity worth memorizing:

\[\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b\]

So if you know one, you can find the other. Example: \(a = 12\), \(b = 18\). We computed GCF = 6 and LCM = 36. Check: \(6 \times 36 = 216 = 12 \times 18\). ✓

This identity lets you solve test problems faster. If a question asks for the LCM but you can quickly see the GCF, just divide \(\frac{a \times b}{\text{GCF}}\).

Three-method comparison

Here’s when to use each method:

• Listing factors/multiples. Best for small numbers (under 30). Fast to write, easy for kids.
• Prime factorization. Best for medium numbers (30 to 300) or when factoring more than two numbers. Most reliable.
• Ladder/cake method. Best when speed matters. Once you’ve practiced, it’s the fastest of the three.

Know all three. The test might present the problem in a way that favors one over the others.

Practice set (try these on paper)

1. GCF(48, 60)
2. LCM(8, 12)
3. GCF(24, 36, 60)
4. LCM(6, 10, 15)
5. Two numbers have GCF 4 and product 96. Find them.
6. Bus A every 15 min, bus B every 25 min, next meet time?
7. Greatest number of identical bouquets from 24 roses and 36 tulips?
8. Find LCM(7, 9). (Hint: coprime!)

Answers: 1) 12, 2) 24, 3) 12, 4) 30, 5) 8 and 12, 6) 75 min later, 7) 12 bouquets, 8) 63.

One last shortcut

When two numbers are coprime (their GCF is 1), the LCM is simply their product. This is true for any two consecutive integers, any two primes, and many other pairs. Recognize coprime pairs and you save time.

Extra study tips that move the needle

Most students don’t fail because the math is too hard. They fail because their practice habits are inefficient. Here are the habits that separate the students who improve fast from those who stall.

Practice with a timer. Untimed practice teaches you to eventually get the right answer; timed practice teaches you to get it in test conditions. Set a stopwatch every time you sit down. Aim for 90 seconds per question on most standardized tests.

Keep an error log. A simple spreadsheet with three columns, Problem, My answer, Correct answer, Why I missed it, is the single most powerful study tool ever invented. Review your error log weekly. The same mistakes show up again and again until you name them.

Mix topics every session. Doing 20 problems on the same topic feels productive, but spaced and interleaved practice, mixing topics, builds retrieval skills, which is what the test actually measures. Spend 70% of your time on mixed sets and only 30% on isolated drills.

Sleep on it. Memory consolidation happens during sleep. A 30-minute session the night before a quiz, followed by 7+ hours of sleep, beats a 3-hour cram session that ends at midnight. This is settled cognitive science.

Teach the topic out loud. If you can’t explain it, you don’t fully know it. Either record yourself, write a one-paragraph “how I’d teach this” explanation, or grab a friend to listen. Teaching exposes the gaps your problem sets hid.

When to ask for help

Spinning your wheels for more than 15 minutes on a single problem is a signal, not of failure, but of a missing piece of background. Stop, mark the problem, and either ask a teacher, post in our community, or watch a video on the relevant subtopic. Resuming after gaining the missing piece is much more efficient than guessing your way forward.

A quick self-assessment

Before you close this tab, answer these three questions honestly:

1. What’s the one topic in this article you understood best?
2. What’s the one topic that still feels fuzzy?
3. What concrete next step (a worksheet, a practice test, a video) will you take in the next 48 hours?

Writing those answers down, even just in a notes app, has been shown to roughly double the chance you actually follow through. Treat the next 48 hours as a small, doable experiment, not a marathon. Your future test-day self will thank you.

Free practice in our 5th- and 6th-grade worksheets, and a full review in the Middle School Bundle.