The remainder when `(12^(13) + 23^(13))` is divided by 11 :

Options

A. 0
B. 1
C. 2
D. None of these

Answer

To find the remainder when \( 12^{13} + 23^{13} \) is divided by 11, we can follow these steps:

### Step 1: Simplify the bases modulo 11
First, we need to find \( 12 \mod 11 \) and \( 23 \mod 11 \).

- \( 12 \mod 11 = 1 \) (since \( 12 - 11 = 1 \))
- \( 23 \mod 11 = 1 \) (since \( 23 - 2 \times 11 = 1 \))

### Step 2: Rewrite the expression
Now we can rewrite the expression \( 12^{13} + 23^{13} \) using the results from Step 1:

\[ 12^{13} + 23^{13} \equiv 1^{13} + 1^{13} \mod 11 \]

### Step 3: Calculate the powers
Next, we calculate \( 1^{13} \):

\[ 1^{13} = 1 \]

So, we have:

\[ 1^{13} + 1^{13} = 1 + 1 = 2 \]

### Step 4: Find the remainder
Now, we find the remainder when \( 2 \) is divided by \( 11 \):

\[ 2 \mod 11 = 2 \]

Thus, the remainder when \( 12^{13} + 23^{13} \) is divided by \( 11 \) is \( 2 \).

### Final Answer
The remainder is \( 2 \).

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