The LCM of 18 and 24 is 72. This guide covers everything you need to know about finding the least common multiple of 18 and 24: three proven methods, step-by-step examples, visual Venn diagrams, the relationship between LCM and GCD, real-world applications, and answers to frequently asked questions.
The LCM of 18 and 24 is 72. The Least Common Multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers without leaving a remainder.
For 18 and 24, the LCM is 72 because:
Visualization of the multiples of 18 and 24 on a common number scale. The very first point where both lines hit the exact same value is 72.
Easily understand how prime factors combine: the common factors form the GCD (6), and all combined factors form the LCM (72).
There are three standard methods to find the LCM of 18 and 24. Each method gives the same result: 72.
List the multiples of 18 and 24 and find the smallest number they share.
Break 18 and 24 into prime factors, take the highest power of each prime, and multiply.
Divide 18 and 24 by common prime factors until no further division is possible.
The prime factorization method is the most reliable way to find the LCM of 18 and 24.
| Number | Prime Factorization | 2 | 3 | Highest Power |
|---|---|---|---|---|
| 18 | 2 × 32 | 21 | 32 | — |
| 24 | 23 × 3 | 23 | 31 | — |
| LCM | 23 × 32 | 23 | 32 | 8 × 9 = 72 |
In the division method, we divide 18 and 24 by prime factors until no common factor other than 1 remains.
| Prime Factor | 18 | 24 |
|---|---|---|
| 2 | 9 | 12 |
| 2 | 9 | 6 |
| 2 | 9 | 3 |
| 3 | 3 | 1 |
| 3 | 1 | 1 |
LCM(18, 24) = 2 × 2 × 2 × 3 × 3 = 72
List the multiples of 18 and 24 until you find the smallest common multiple.
| Number | Multiples | First Common |
|---|---|---|
| 18 | 18, 36, 54, 72, 90, 108, … | 72 |
| 24 | 24, 48, 72, 96, 120, 144, … |
The LCM and GCD (Greatest Common Divisor) of 18 and 24 are different values that serve complementary purposes in mathematics.
| Aspect | LCM of 18 and 24 | GCD of 18 and 24 |
|---|---|---|
| Definition | Smallest number divisible by both 18 and 24 | Largest number that divides both 18 and 24 |
| Value | 72 | 6 |
| Relationship | LCM(18, 24) × GCD(18, 24) = 72 × 6 = 432 = 18 × 24 ✓ | |
| Size | Always ≥ 24 (the largest input) | Always ≤ 18 (the smallest input) |
| Primary Use | Adding fractions, scheduling | Simplifying fractions, grouping |
The LCM of 18 and 24 isn't just a mathematical concept — it appears in many practical situations where repeating events need to align.
| Field | Problem | LCM Role | Example |
|---|---|---|---|
| Mathematics | Adding fractions | Find common denominator | 1/18 + 1/24 = 4/72 + 3/72 = 7/72 |
| Scheduling | Repeating events | When events align | Bus every 18 min & 24 min → every 72 min |
| Mechanics | Gear rotation | Re-alignment point | 18-tooth & 24-tooth gears → 72 turns |
| Commerce | Package sizes | Equal quantities | 18-pack & 24-pack → buy 72 of each |
| Music | Rhythm pattern | Pattern re-sync | 18/8 & 24/8 time → 72 beats |
Visual explanation of why LCM is crucial in mechanics: Gear A (18 teeth) and Gear B (24 teeth) align at tooth 72 after Gear A makes 4 full rotations and Gear B makes 3 full rotations.
To add 1/18 + 1/24, find the LCD (Least Common Denominator), which is LCM(18, 24) = 72.
If one bus leaves every 18 minutes and another every 24 minutes, they will depart together every LCM(18, 24) = 72 minutes (1 hour 12 minutes).
Three fully solved examples using different methods.
| Step | Action | Result |
|---|---|---|
| 1 | Prime factorize 18 and 24 | 18 = 2 × 3²; 24 = 2³ × 3 |
| 2 | Take highest powers | 2³, 3² |
| 3 | Multiply | 8 × 9 = 72 |
| ✓ | Verify | 72 ÷ 18 = 4 ✓; 72 ÷ 24 = 3 ✓ |
Find LCM(18, 24) using GCD(18, 24) = 6.
Find the first common multiple of 18 and 24.
Even experienced students make these errors. Watch out for them.
| ❌ Mistake | Why it's wrong | ✅ Correct approach |
|---|---|---|
| Saying the LCM is 144 | 144 is a common multiple, but not the least common multiple | The LCM of 18 and 24 is 72 |
| Confusing LCM with GCD | GCD(18, 24) = 6, but LCM(18, 24) = 72 | Remember: LCM ≥ max(18, 24); GCD ≤ min(18, 24) |
| Multiplying 18 × 24 = 432 | Only works when numbers are coprime (GCD = 1). GCD(18, 24) = 6 | Use LCM = (18 × 24) ÷ GCD = 432 ÷ 6 = 72 |
| Forgetting the highest power | Using 3¹ instead of 3² gives the wrong result | Always pick the maximum exponent for each prime |
Quick answers to the most common questions about the LCM of 18 and 24.
The LCM of 18 and 24 is 72. It is the smallest positive integer that is divisible by both 18 and 24.
You can find the LCM of 18 and 24 using three methods: (1) Prime Factorization — 18 = 2 × 3² and 24 = 2³ × 3, so LCM = 2³ × 3² = 72; (2) Division Method — divide by prime factors; or (3) Listing Multiples — list multiples until you find 72.
The LCM of 18, 24, and 36 is 72. Prime factorization: 18 = 2 × 3², 24 = 2³ × 3, 36 = 2² × 3². Highest powers: 2³ × 3² = 8 × 9 = 72.
No. While 144 is a common multiple of 18 and 24, it is not the least common multiple. The LCM of 18 and 24 is 72. 144 is simply the second common multiple.
The GCD (Greatest Common Divisor) of 18 and 24 is 6. The common factors of 18 and 24 are 1, 2, 3, and 6, and 6 is the largest.
For any two numbers: LCM(a, b) × GCD(a, b) = a × b. For 18 and 24: LCM(18, 24) × GCD(18, 24) = 72 × 6 = 432 = 18 × 24 ✓
LCM(18, 24) = 72 is used when adding fractions with denominators 18 and 24, when scheduling events that repeat every 18 and 24 minutes, and when finding when two gears with 18 and 24 teeth will realign.
The LCM of 18 and 24 is 72, which is an integer. The formula for LCM of fractions is different: LCM(a₁/b₁, a₂/b₂) = LCM(numerators) ÷ GCD(denominators).
Key terms related to the LCM of 18 and 24.