The remainder of `((36)^(13))/7`?

Options

A. 1
B. 6
C. 2
D. 5

Answer

To find the remainder of \( \frac{36^{13}}{7} \), we can use properties of modular arithmetic. Here’s a step-by-step solution:

### Step 1: Simplify the base modulo 7
First, we find \( 36 \mod 7 \):
\[ 36 \div 7 = 5 \quad \text{(since } 7 \times 5 = 35\text{)} \]
\[ 36 - 35 = 1 \quad \Rightarrow \quad 36 \equiv 1 \mod 7 \]

### Step 2: Substitute the simplified base into the exponent
Now, we can replace \( 36 \) with \( 1 \) in our expression:
\[ 36^{13} \equiv 1^{13} \mod 7 \]

### Step 3: Calculate the power
Since any number raised to a power of 1 is still 1:
\[ 1^{13} = 1 \]

### Step 4: Find the remainder
Now we find the remainder when \( 1 \) is divided by \( 7 \):
\[ 1 \mod 7 = 1 \]

### Conclusion
Thus, the remainder of \( \frac{36^{13}}{7} \) is:
\[ \boxed{1} \]

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