The LCM of 24 and 36 is 72. This guide covers everything you need to know about finding the least common multiple of 24 and 36: three proven methods, interactive site-themed SVG Venn and gear synchronization diagrams, step-by-step examples, the relationship between LCM and GCD, real-world applications, and answers to frequently asked questions.
The LCM of 24 and 36 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 24 and 36, the LCM is 72 because:
Visualization of the multiples of 24 and 36 on a number line using the brand color palette. Both series meet at 72.
There are three standard methods to find the LCM of 24 and 36. Each method gives the exact same result: 72.
List the multiples of 24 and 36 and find the smallest number they share.
Break 24 and 36 into prime factors, take the highest power of each prime, and multiply.
Divide 24 and 36 by common prime factors until no further division is possible.
The prime factorization method is the most reliable way to find the LCM of 24 and 36.
Designed with website theme colors: The overlapping intersection contains the common prime factors 2 × 2 × 3 = 12 (GCD), while the union of all factors (2³ × 3² = 72) gives the LCM.
| Number | Prime Factorization | 2 | 3 | Highest Power |
|---|---|---|---|---|
| 24 | 23 × 3 | 23 | 31 | — |
| 36 | 22 × 32 | 22 | 32 | — |
| LCM | 23 × 32 | 23 | 32 | 8 × 9 = 72 |
In the division method, we divide 24 and 36 by prime factors until no common factor other than 1 remains.
| Prime Factor | 24 | 36 |
|---|---|---|
| 2 | 12 | 18 |
| 2 | 6 | 9 |
| 3 | 2 | 3 |
| 2 | 1 | 3 |
| 3 | 1 | 1 |
LCM(24, 36) = 2 × 2 × 3 × 2 × 3 = 72
List the multiples of 24 and 36 until you find the smallest common multiple.
| Number | Multiples | First Common |
|---|---|---|
| 24 | 24, 48, 72, 96, 120, 144, … | 72 |
| 36 | 36, 72, 108, 144, 180, … |
The LCM and GCD (Greatest Common Divisor) of 24 and 36 are different values that serve complementary purposes in mathematics.
| Aspect | LCM of 24 and 36 | GCD of 24 and 36 |
|---|---|---|
| Definition | Smallest number divisible by both 24 and 36 | Largest number that divides both 24 and 36 |
| Value | 72 | 12 |
| Relationship | LCM(24, 36) × GCD(24, 36) = 72 × 12 = 864 = 24 × 36 ✓ | |
| Size | Always ≥ 36 (the largest input) | Always ≤ 24 (the smallest input) |
| Primary Use | Adding fractions, scheduling | Simplifying fractions, grouping |
The LCM of 24 and 36 isn't just a mathematical concept — it appears in many practical situations where repeating events need to align.
Rendered with site brand colors: A 24-tooth gear completes 3 full revolutions, while a 36-tooth gear completes 2 full revolutions to re-synchronize at 72 units.
| Field | Problem | LCM Role | Example |
|---|---|---|---|
| Mathematics | Adding fractions | Find common denominator | 1/24 + 1/36 = 3/72 + 2/72 = 5/72 |
| Scheduling | Repeating events | When events align | Bus every 24 min & 36 min → every 72 min |
| Mechanics | Gear rotation | Re-alignment point | 24-tooth & 36-tooth gears → 72 turns |
| Commerce | Package sizes | Equal quantities | 24-pack & 36-pack → buy 72 of each |
| Music | Rhythm pattern | Pattern re-sync | 24/8 & 36/8 time → 72 beats |
To add 1/24 + 1/36, find the LCD (Least Common Denominator), which is LCM(24, 36) = 72.
If one bus leaves every 24 minutes and another every 36 minutes, they will depart together every LCM(24, 36) = 72 minutes.
Three fully solved examples using different methods.
| Step | Action | Result |
|---|---|---|
| 1 | Prime factorize 24 and 36 | 24 = 2³ × 3; 36 = 2² × 3² |
| 2 | Take highest powers | 2³, 3² |
| 3 | Multiply | 8 × 9 = 72 |
| ✓ | Verify | 72 ÷ 24 = 3 ✓; 72 ÷ 36 = 2 ✓ |
Find LCM(24, 36) using GCD(24, 36) = 12.
Find the first common multiple of 24 and 36.
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 24 and 36 is 72 |
| Confusing LCM with GCD | GCD(24, 36) = 12, but LCM(24, 36) = 72 | Remember: LCM ≥ max(24, 36); GCD ≤ min(24, 36) |
| Multiplying 24 × 36 = 864 | Only works when numbers are coprime (GCD = 1). GCD(24, 36) = 12 | Use LCM = (24 × 36) ÷ GCD = 864 ÷ 12 = 72 |
| Forgetting the highest power | Using 2² instead of 2³ gives the wrong result | Always pick the maximum exponent for each prime |
Quick answers to the most common questions about the LCM of 24 and 36.
The LCM of 24 and 36 is 72. It is the smallest positive integer that is divisible by both 24 and 36.
You can find the LCM of 24 and 36 using three methods: (1) Prime Factorization — 24 = 2³ × 3 and 36 = 2² × 3², 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 24, 36, and 48 is 144. Prime factorization: 24 = 2³ × 3, 36 = 2² × 3², 48 = 2⁴ × 3. Highest powers: 2⁴ × 3² = 16 × 9 = 144.
No. While 144 is a common multiple of 24 and 36, it is not the least common multiple. The LCM of 24 and 36 is 72. 144 is simply the second common multiple.
The GCD (Greatest Common Divisor) of 24 and 36 is 12. The common factors of 24 and 36 are 1, 2, 3, 4, 6, and 12, and 12 is the largest.
For any two numbers: LCM(a, b) × GCD(a, b) = a × b. For 24 and 36: LCM(24, 36) × GCD(24, 36) = 72 × 12 = 864 = 24 × 36 ✓
LCM(24, 36) = 72 is used when adding fractions with denominators 24 and 36, when scheduling events that repeat every 24 and 36 minutes, and when finding when two gears with 24 and 36 teeth will realign.
The LCM of 24 and 36 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 24 and 36.