The Relationship Between GCF and LCM
The GCF and LCM are not independent — they are intimately connected through a simple but powerful formula. Understanding this connection allows you to find one whenever you know the other.
The Key Formula
For any two positive integers a and b: GCF(a, b) × LCM(a, b) = a × b
Proving It with an Example
GCF(12,18) = 6. LCM(12,18) = 36.
6 × 36 = 216 = 12 × 18 ✓
Using the Formula Both Ways
GCF × LCM = 20 × ? — we need more information to find both numbers, but if we also know LCM = 60: other number = GCF × LCM ÷ 20 = 4 × 60 ÷ 20 = 12.
Why Does It Work? Venn Diagram Insight
Think of prime factors placed in a Venn diagram with two circles. GCF uses only the intersection (shared primes at lowest power). LCM uses the entire union (all primes at highest power). The product of GCF × LCM counts every prime factor the same number of times as in a × b.
Important Note – Three Numbers
The formula GCF × LCM = a × b applies to exactly two numbers. For three or more numbers, there is no direct equivalent — use prime factorization instead.
Summary Table
| Quantity | Uses Primes at | Venn Area |
|---|---|---|
| GCF | Lowest power in both | Intersection only |
| LCM | Highest power in either | Full union |
| a × b | All primes in both | Both circles (counted twice) |
Key Takeaways
- GCF × LCM = product of the two numbers (for two numbers only).
- If you know GCF, find LCM as (a × b) ÷ GCF and vice versa.
- GCF uses intersection of prime factors; LCM uses the union.
Practice Questions
- GCF(8,12) = 4. Find LCM(8,12) without listing multiples.
- LCM(6,10) = 30. Find GCF(6,10).
- Verify the formula for 15 and 25.
- Two numbers have GCF 6 and LCM 60. One number is 12. Find the other.
- Can the GCF of two numbers ever equal their LCM? When?