`1023^(26)` is divisible by

Options

A. 3
B. 9 & 3
C. 11
D. 3, 9 & 11

Answer

To determine what \( 1023^{26} \) is divisible by, we will first factor \( 1023 \) and then analyze its prime factors raised to the power of \( 26 \).

### Step 1: Factor \( 1023 \)
We start by finding the prime factorization of \( 1023 \).

1. Check divisibility by \( 3 \):
- The sum of the digits of \( 1023 \) is \( 1 + 0 + 2 + 3 = 6 \), which is divisible by \( 3 \).
- Dividing \( 1023 \) by \( 3 \):
\[ 1023 \div 3 = 341 \]

2. Now factor \( 341 \):
- Check divisibility by \( 11 \):
- The alternating sum is \( 3 - 4 + 1 = 0 \), which is divisible by \( 11 \).
- Dividing \( 341 \) by \( 11 \):
\[ 341 \div 11 = 31 \]

3. Now we have \( 31 \), which is a prime number.

Thus, the complete factorization of \( 1023 \) is:
\[ 1023 = 3^1 \times 11^1 \times 31^1 \]

### Step 2: Raise the factors to the power of \( 26 \)
Now we raise each factor to the power of \( 26 \):
\[ 1023^{26} = (3^1 \times 11^1 \times 31^1)^{26} = 3^{26} \times 11^{26} \times 31^{26} \]

### Step 3: Determine divisibility
Now we check the divisibility of \( 1023^{26} \) by the numbers given in the options: \( 3 \), \( 9 \), and \( 11 \).

1. **Divisibility by \( 3 \)**:
- Since \( 1023^{26} \) includes \( 3^{26} \), it is clearly divisible by \( 3 \).

2. **Divisibility by \( 9 \)**:
- \( 9 = 3^2 \). Since \( 1023^{26} \) includes \( 3^{26} \), it is also divisible by \( 9 \) because \( 26 \geq 2 \).

3. **Divisibility by \( 11 \)**:
- Since \( 1023^{26} \) includes \( 11^{26} \), it is clearly divisible by \( 11 \).

### Conclusion
Thus, \( 1023^{26} \) is divisible by \( 3 \), \( 9 \), and \( 11 \).

### Final Answer
The answer is that \( 1023^{26} \) is divisible by \( 3, 9, \) and \( 11 \).

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