Side-by-Side Example Using 18 and 24
Let's put 18 and 24 side by side and compute both values from the exact same starting numbers:
Factors = Small Pieces Inside
GCF(18, 24) = 6
What it asks: What is the largest whole number that divides into both 18 and 24 evenly?
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Largest match = 6
Prime rule: Keep only shared primes at their lowest power (2¹ × 3¹ = 6).
Multiples = Larger Products Ahead
LCM(18, 24) = 72
What it asks: What is the smallest future number that both 18 and 24 can grow into by multiplication?
Multiples of 18: 18, 36, 54, 72, 90…
Multiples of 24: 24, 48, 72, 96…
Smallest match = 72
Prime rule: Keep every prime at its highest power (2³ × 3² = 72).
How to Tell Whether a Word Problem Wants LCM or GCF
When a test question doesn't explicitly say “find the LCM” or “find the GCF,” look at what you are doing to the quantities:
| Clue in the Story | Use LCM | Use GCF |
|---|---|---|
| Action being taken | Repeating cycles, buying multiple packs to match, future events aligning | Cutting ribbon into equal pieces, dividing students into identical teams, packing gift bags |
| Size of the answer | As large as or larger than the biggest number | As small as or smaller than the smallest number |
| Typical question phrasing | “When will both happen together again?” or “What is the smallest square…” | “What is the greatest number of identical groups you can make with none left over?” |
| Use with fractions | Finding a common denominator (LCD) to add or subtract | Dividing top and bottom to simplify/reduce a fraction to lowest terms |
The Golden Equation Connecting Both
For any two numbers a and b, multiplying their GCF and LCM gives the exact same product as multiplying the two numbers themselves:
GCF(18, 24) × LCM(18, 24) = 6 × 72 = 432 = 18 × 24