Find the remainder when `7^(187)` is divided by 800.

Options

A. 143
B. 243
C. 343
D. 443

Answer

To find the remainder when \( 7^{187} \) is divided by 800, we can use modular arithmetic and properties of exponents. Here’s a step-by-step solution:

### Step 1: Break down the exponent
We can express \( 7^{187} \) in a more manageable form. Notice that \( 187 \) can be expressed as \( 4 \times 46 + 3 \):
\[ 7^{187} = 7^{4 \times 46 + 3} = (7^4)^{46} \cdot 7^3 \]

### Step 2: Calculate \( 7^4 \)
Now, we calculate \( 7^4 \):
\[ 7^4 = 2401 \]

### Step 3: Find \( 2401 \mod 800 \)
Next, we need to find \( 2401 \mod 800 \):
\[ 2401 \div 800 = 3.00125 \quad \text{(which means 3 full times)} \]
Calculating the product:
\[ 3 \times 800 = 2400 \]
Now, subtract this from 2401 to find the remainder:
\[ 2401 - 2400 = 1 \]
Thus,
\[ 7^4 \equiv 1 \mod 800 \]

### Step 4: Substitute back into the expression
Using the result from Step 3, we substitute back into our expression for \( 7^{187} \):
\[ 7^{187} \equiv (7^4)^{46} \cdot 7^3 \equiv 1^{46} \cdot 7^3 \equiv 7^3 \mod 800 \]

### Step 5: Calculate \( 7^3 \)
Now we calculate \( 7^3 \):
\[ 7^3 = 343 \]

### Step 6: Find the final remainder
Now we need to find \( 343 \mod 800 \):
Since \( 343 < 800 \), the remainder when \( 343 \) is divided by \( 800 \) is simply \( 343 \).

Thus, the remainder when \( 7^{187} \) is divided by \( 800 \) is:
\[ \boxed{343} \]