If the number `23^(32) - 9` is divided by 16, then the remainder is :

Options

A. 8
B. 0
C. 6
D. none of these

Answer

To find the remainder when \( 23^{32} - 9 \) is divided by 16, we can follow these steps:

### Step 1: Rewrite the base
First, we can express \( 23 \) in a form that is easier to work with when dividing by \( 16 \):
\[ 23 \equiv 7 \mod 16 \]
This means that \( 23 \) can be replaced by \( 7 \) when considering division by \( 16 \).

### Step 2: Substitute and simplify
Now, we can substitute \( 23 \) with \( 7 \) in our expression:
\[ 23^{32} - 9 \equiv 7^{32} - 9 \mod 16 \]

### Step 3: Calculate \( 7^{32} \mod 16 \)
Next, we need to find \( 7^{32} \mod 16 \). We can use the property of powers and modular arithmetic. First, we can calculate the lower powers of \( 7 \):
- \( 7^1 \equiv 7 \mod 16 \)
- \( 7^2 = 49 \equiv 1 \mod 16 \)

Since \( 7^2 \equiv 1 \mod 16 \), we can conclude that:
\[ 7^{32} = (7^2)^{16} \equiv 1^{16} \equiv 1 \mod 16 \]

### Step 4: Substitute back into the expression
Now we can substitute \( 7^{32} \) back into our expression:
\[ 7^{32} - 9 \equiv 1 - 9 \mod 16 \]

### Step 5: Simplify further
Now we simplify \( 1 - 9 \):
\[ 1 - 9 = -8 \]
To find the positive remainder, we can add \( 16 \):
\[ -8 + 16 = 8 \]

### Conclusion
Thus, the remainder when \( 23^{32} - 9 \) is divided by \( 16 \) is:
\[ \boxed{8} \]