The remainder when `7^(84)` is divided by 342 is

Options

A. a) 0
B. b) 1
C. c) 21
D. d) 340

Answer

To find the remainder when \( 7^{84} \) is divided by \( 342 \), we can follow these steps:

### Step 1: Factor the denominator
First, we factor \( 342 \) to see if we can simplify our calculations:
\[ 342 = 2 \times 171 = 2 \times 3 \times 57 = 2 \times 3 \times 3 \times 19 = 2 \times 3^2 \times 19 \]

### Step 2: Use the property of powers
Notice that \( 7^3 = 343 \). We can express \( 7^{84} \) as:
\[ 7^{84} = (7^3)^{28} = 343^{28} \]

### Step 3: Calculate the remainder
Now, we need to find the remainder when \( 343^{28} \) is divided by \( 342 \). Since \( 343 \) is \( 1 \) more than \( 342 \):
\[ 343 \equiv 1 \mod 342 \]
Thus,
\[ 343^{28} \equiv 1^{28} \mod 342 \]
This simplifies to:
\[ 1^{28} = 1 \]

### Step 4: Conclusion
Therefore, the remainder when \( 7^{84} \) is divided by \( 342 \) is:
\[ \text{Remainder} = 1 \]

### Final Answer
The remainder when \( 7^{84} \) is divided by \( 342 \) is \( 1 \).

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