What is the least common multiple of 24, 48 and 72?

Options

A. A)72
B. B)144
C. C)288
D. D)216

Answer

To find the least common multiple (LCM) of the numbers 24, 48, and 72, we can follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.

- **For 24**:
- 24 can be divided by 2: \( 24 \div 2 = 12 \)
- 12 can be divided by 2: \( 12 \div 2 = 6 \)
- 6 can be divided by 2: \( 6 \div 2 = 3 \)
- 3 is a prime number.

Therefore, the prime factorization of 24 is:
\[ 24 = 2^3 \times 3^1 \]

- **For 48**:
- 48 can be divided by 2: \( 48 \div 2 = 24 \)
- 24 can be divided by 2: \( 24 \div 2 = 12 \)
- 12 can be divided by 2: \( 12 \div 2 = 6 \)
- 6 can be divided by 2: \( 6 \div 2 = 3 \)
- 3 is a prime number.

Therefore, the prime factorization of 48 is:
\[ 48 = 2^4 \times 3^1 \]

- **For 72**:
- 72 can be divided by 2: \( 72 \div 2 = 36 \)
- 36 can be divided by 2: \( 36 \div 2 = 18 \)
- 18 can be divided by 2: \( 18 \div 2 = 9 \)
- 9 can be divided by 3: \( 9 \div 3 = 3 \)
- 3 is a prime number.

Therefore, the prime factorization of 72 is:
\[ 72 = 2^3 \times 3^2 \]

### Step 2: Identify the Highest Powers
Next, we identify the highest powers of each prime factor from the factorizations:

- For the prime number 2:
- Highest power from 24: \( 2^3 \)
- Highest power from 48: \( 2^4 \)
- Highest power from 72: \( 2^3 \)

The highest power is \( 2^4 \).

- For the prime number 3:
- Highest power from 24: \( 3^1 \)
- Highest power from 48: \( 3^1 \)
- Highest power from 72: \( 3^2 \)

The highest power is \( 3^2 \).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying the highest powers of all prime factors together:
\[ \text{LCM} = 2^4 \times 3^2 \]

Calculating this:
- \( 2^4 = 16 \)
- \( 3^2 = 9 \)

Now, multiply these results:
\[ \text{LCM} = 16 \times 9 = 144 \]

### Conclusion
Thus, the least common multiple of 24, 48, and 72 is:
\[ \text{LCM} = 144 \]

### Answer
The correct option is **B) 144**.