The LCM of 16 and 24 is 48. This guide covers everything you need to know about finding the least common multiple of 16 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 16 and 24 is 48. The Least Common Multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers without leaving a remainder.
For 16 and 24, the LCM is 48 because:
Visualization of the first 10 multiples of 16 and 24. The first common multiple where both numbers meet is 48.
Easily understand how prime factors combine: the common factors 2 × 2 × 2 form the GCD (8), while multiplying all prime factors in the union gives the LCM (48).
There are three standard methods to find the LCM of 16 and 24. Each method gives the same result: 48.
List the multiples of 16 and 24 and find the smallest number they share.
Break 16 and 24 into prime factors, take the highest power of each prime, and multiply.
Divide 16 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 16 and 24.
| Number | Prime Factorization | 2 | 3 | Highest Power |
|---|---|---|---|---|
| 16 | 24 | 24 | — | — |
| 24 | 23 × 3 | 23 | 31 | — |
| LCM | 24 × 31 | 24 | 31 | 16 × 3 = 48 |
In the division method, we divide 16 and 24 by prime factors until no common factor other than 1 remains.
| Prime Factor | 16 | 24 |
|---|---|---|
| 2 | 8 | 12 |
| 2 | 4 | 6 |
| 2 | 2 | 3 |
| 2 | 1 | 3 |
| 3 | 1 | 1 |
LCM(16, 24) = 2 × 2 × 2 × 2 × 3 = 48
List the multiples of 16 and 24 until you find the smallest common multiple.
| Number | Multiples | First Common |
|---|---|---|
| 16 | 16, 32, 48, 64, 80, 96, … | 48 |
| 24 | 24, 48, 72, 96, 120, 144, … |
The LCM and GCD (Greatest Common Divisor) of 16 and 24 are different values that serve complementary purposes in mathematics.
| Aspect | LCM of 16 and 24 | GCD of 16 and 24 |
|---|---|---|
| Definition | Smallest number divisible by both 16 and 24 | Largest number that divides both 16 and 24 |
| Value | 48 | 8 |
| Relationship | LCM(16, 24) × GCD(16, 24) = 48 × 8 = 384 = 16 × 24 ✓ | |
| Size | Always ≥ 24 (the largest input) | Always ≤ 16 (the smallest input) |
| Primary Use | Adding fractions, scheduling | Simplifying fractions, grouping |
The LCM of 16 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 | 1/16 + 1/24 = 3/48 + 2/48 = 5/48 |
| Scheduling | Repeating events | When events align | Bus every 16 min & 24 min → every 48 min |
| Mechanics | Gear rotation | Re-alignment point | 16-tooth & 24-tooth gears → 48 turns |
| Commerce | Package sizes | Equal quantities | 16-pack & 24-pack → buy 48 of each |
| Music | Rhythm pattern | Pattern re-sync | 16/8 & 24/8 time → 48 beats |
Visual explanation of alignment: A 16-tooth gear rotates exactly 3 times while a 24-tooth gear rotates exactly 2 times to align at tooth 48. Buses departing every 16 and 24 minutes meet at minute 48.
To add 1/16 + 1/24, find the LCD (Least Common Denominator), which is LCM(16, 24) = 48.
If one bus leaves every 16 minutes and another every 24 minutes, they will depart together every LCM(16, 24) = 48 minutes.
Three fully solved examples using different methods.
| Step | Action | Result |
|---|---|---|
| 1 | Prime factorize 16 and 24 | 16 = 2⁴; 24 = 2³ × 3 |
| 2 | Take highest powers | 2⁴, 3¹ |
| 3 | Multiply | 16 × 3 = 48 |
| ✓ | Verify | 48 ÷ 16 = 3 ✓; 48 ÷ 24 = 2 ✓ |
Find LCM(16, 24) using GCD(16, 24) = 8.
Find the first common multiple of 16 and 24.
Even experienced students make these errors. Watch out for them.
| ❌ Mistake | Why it's wrong | ✅ Correct approach |
|---|---|---|
| Saying the LCM is 96 | 96 is a common multiple, but not the least common multiple | The LCM of 16 and 24 is 48 |
| Confusing LCM with GCD | GCD(16, 24) = 8, but LCM(16, 24) = 48 | Remember: LCM ≥ max(16, 24); GCD ≤ min(16, 24) |
| Multiplying 16 × 24 = 384 | Only works when numbers are coprime (GCD = 1). GCD(16, 24) = 8 | Use LCM = (16 × 24) ÷ GCD = 384 ÷ 8 = 48 |
| Forgetting the highest power | Using 2³ instead of 2⁴ gives the wrong result | Always pick the maximum exponent for each prime |
Quick answers to the most common questions about the LCM of 16 and 24.
The LCM of 16 and 24 is 48. It is the smallest positive integer that is divisible by both 16 and 24.
You can find the LCM of 16 and 24 using three methods: (1) Prime Factorization — 16 = 2⁴ and 24 = 2³ × 3, so LCM = 2⁴ × 3 = 48; (2) Division Method — divide by prime factors; or (3) Listing Multiples — list multiples until you find 48.
The LCM of 16, 24, and 48 is 48. Since 48 is already a multiple of both 16 and 24, the LCM remains 48.
No. While 96 is a common multiple of 16 and 24, it is not the least common multiple. The LCM of 16 and 24 is 48. 96 is simply the second common multiple.
The GCD (Greatest Common Divisor) of 16 and 24 is 8. The common factors of 16 and 24 are 1, 2, 4, and 8, and 8 is the largest.
For any two numbers: LCM(a, b) × GCD(a, b) = a × b. For 16 and 24: LCM(16, 24) × GCD(16, 24) = 48 × 8 = 384 = 16 × 24 ✓
LCM(16, 24) = 48 is used when adding fractions with denominators 16 and 24, when scheduling events that repeat every 16 and 24 minutes, and when finding when two gears with 16 and 24 teeth will realign.
The LCM of 16 and 24 is 48, 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 16 and 24.