Pre-Algebra
Numbers and operations
GCF and LCM
20 practice questions
0 video lessons
Theory + worked examples
GCF and LCM
California Pre-Algebra • Standard 6.NS.4 • Numbers & Operations
GCF and LCM is a topic in Numbers & Operations in the California Common Core State Standards. It is aligned to Standard 6.NS.4, which requires students to find the greatest common factor and least common multiple of whole numbers.
The GCF is the largest factor two numbers share and the LCM is the smallest multiple they share.
Theory
The greatest common factor (GCF) is the largest number dividing both; the least common multiple (LCM) is the smallest number both divide.
Prime factorization makes both quick to find.
GCF and LCM.
Using prime factors.
A useful identity:
\[\gcd(a,b)\cdot\operatorname{lcm}(a,b)=a\cdot b\]
GCF for simplifying, LCM for common denominators.
How to find GCF and LCM
- Factor each number into primes.
- GCF: multiply the shared prime powers.
- LCM: multiply the highest power of each prime.
- Check with the product identity.
Example 1 — GCF
Find \(\gcd(12,18)\).
Solution
Largest factor in both.
| \(\gcd(12,18)\) | \(=\) | \(6\) |
Example 2 — LCM
Find \(\operatorname{lcm}(4,6)\).
Solution
Smallest number both divide.
| \(\operatorname{lcm}(4,6)\) | \(=\) | \(12\) |
Example 3 — Prime factors
Use primes: \(\gcd(24,36)\).
Solution
\(24=2^3\cdot3\), \(36=2^2\cdot3^2\); shared \(2^2\cdot3\).
| \(\gcd\) | \(=\) | \(12\) |
Example 4 — Word problem
Hot dogs come in \(10\), buns in \(8\). Fewest to match?
Solution
Use the LCM.
| \(\operatorname{lcm}(10,8)\) | \(=\) | \(40\) |
Common pitfalls
GCF \(\le\) each number; LCM \(\ge\) each number.
Don't swap them — factor vs multiple.
Use the LCM for common denominators.
Frequently asked questions
What is the GCF?
The largest factor two numbers share.
What is the LCM?
The smallest multiple two numbers share.
What is \(\gcd(12,18)\)?
\(6\).
What is \(\operatorname{lcm}(4,6)\)?
\(12\).
More in Numbers and operations