`2^(10) -1` is divisible by

Options

A. 4
B. 10
C. 2
D. 3

Answer

To determine what \( 2^{10} - 1 \) is divisible by, we can follow these steps:

### Step 1: Calculate \( 2^{10} \)
First, we need to calculate \( 2^{10} \):
\[ 2^{10} = 1024 \]

### Step 2: Subtract 1 from \( 2^{10} \)
Now, we subtract 1 from the result:
\[ 2^{10} - 1 = 1024 - 1 = 1023 \]

### Step 3: Determine the divisibility of 1023
Next, we need to check the divisibility of 1023 by the given options: 2, 3, 4, and 10.

1. **Check divisibility by 2**:
- A number is divisible by 2 if it is even. Since 1023 is odd, it is **not divisible by 2**.

2. **Check divisibility by 3**:
- A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 1023 is \( 1 + 0 + 2 + 3 = 6 \), which is divisible by 3. Therefore, **1023 is divisible by 3**.

3. **Check divisibility by 4**:
- A number is divisible by 4 if the last two digits form a number that is divisible by 4. The last two digits of 1023 are 23, and since \( 23 \div 4 = 5.75 \) (not an integer), **1023 is not divisible by 4**.

4. **Check divisibility by 10**:
- A number is divisible by 10 if it ends in 0. Since 1023 ends in 3, it is **not divisible by 10**.

### Conclusion
From the checks above, we find that \( 2^{10} - 1 = 1023 \) is divisible by **3**.