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LCM & GCF Calculator — Free Online Math Tool
Find the Least Common Multiple and Greatest Common Factor with high-precision engineering steps.
4 Numbers detected
account_treeEuclidean Algorithm Breakdown
Finding the GCF of the two largest numbers (18 and 12) using sequential division.
18 = 1 × 12 + 6
Next: GCF(12, 6)
12 = 2 × 6 + 0
Remainder is 0. GCF is 6.
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Frequently Asked Questions
LCM and GCF: The Two Questions About Shared Structure
Which one does your problem need?
They answer opposite questions about the same numbers:
| Problem smells like... | You need | Example |
|---|---|---|
| When do cycles align / repeat together? | LCM | Buses every 12 and 18 min meet every 36 |
| Largest equal grouping / cutting? | GCF | Largest tile fitting 12×18 ft: 6 ft |
| Common denominator for fractions | LCM of denominators | 1/6 + 1/8 → 24ths |
| Simplifying a fraction | GCF of both parts | 18/24 ÷ 6 → 3/4 |
| Splitting items into identical sets | GCF | 36 pens, 48 pencils → 12 identical kits |
The relationship worth remembering
For two numbers, LCM × GCF = the numbers' product — compute one and the other follows. Both are built from prime factorizations: GCF takes the shared primes at lowest powers, LCM takes all primes at highest powers. Seeing 36 = 2²·3² and 48 = 2⁴·3 makes GCF (2²·3 = 12) and LCM (2⁴·3² = 144) readable rather than mysterious.
Built on primes
The factorization underneath comes from the prime factorization tool, and the most common application — fraction arithmetic — runs in the fraction calculator with steps shown.