Identify the GCD of the numbers 32, 64 and 256 among the options.

Options

A. 8
B. 64
C. 16
D. 32

Answer

To find the GCD (Greatest Common Divisor) of the numbers 32, 64, and 256, we can follow these steps:

### Step 1: Prime Factorization of Each Number
- **For 32**:
- 32 can be expressed as \( 2^5 \) (since \( 32 = 2 \times 2 \times 2 \times 2 \times 2 \)).

- **For 64**:
- 64 can be expressed as \( 2^6 \) (since \( 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \)).

- **For 256**:
- 256 can be expressed as \( 2^8 \) (since \( 256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \)).

### Step 2: Identify the Common Factors
Now that we have the prime factorizations:
- \( 32 = 2^5 \)
- \( 64 = 2^6 \)
- \( 256 = 2^8 \)

The common base here is \( 2 \).

### Step 3: Determine the GCD
To find the GCD, we take the lowest power of the common prime factor:
- The lowest power of \( 2 \) among \( 2^5, 2^6, \) and \( 2^8 \) is \( 2^5 \).

### Step 4: Calculate the GCD
Now, we calculate:
- \( 2^5 = 32 \)

Thus, the GCD of 32, 64, and 256 is **32**.

### Conclusion
The GCD of the numbers 32, 64, and 256 is **32**.

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