Identify the number exactly divisible by 12, 18, 72 and 256

Options

A. 4608
B. 1502
C. 2304
D. 751

Answer

To find a number that is exactly divisible by 12, 18, 72, and 256, we will determine the least common multiple (LCM) of these numbers. Here is a step-by-step solution:

### Step 1: Prime Factorization
We start by finding the prime factorization of each number.

- **12**:
- \( 12 = 2^2 \times 3^1 \)

- **18**:
- \( 18 = 2^1 \times 3^2 \)

- **72**:
- \( 72 = 2^3 \times 3^2 \)

- **256**:
- \( 256 = 2^8 \)

### Step 2: Identify the Highest Powers
Next, we identify the highest power of each prime factor that appears in the factorizations.

- For \( 2 \): The highest power is \( 2^8 \) (from 256).
- For \( 3 \): The highest power is \( 3^2 \) (from 18 and 72).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying the highest powers of all prime factors together.

\[ \text{LCM} = 2^8 \times 3^2 \]

### Step 4: Compute the Values
Now, we compute the values:

- \( 2^8 = 256 \)
- \( 3^2 = 9 \)

Now multiply these results:

\[ \text{LCM} = 256 \times 9 \]

Calculating this gives:

\[ 256 \times 9 = 2304 \]

### Conclusion
The least common multiple (LCM) of 12, 18, 72, and 256 is **2304**. Therefore, the number that is exactly divisible by all four numbers is **2304**.

---