Find the remainder when `73xx75xx78xx57xx197` is divided by 34.

Options

A. 22
B. 30
C. 14
D. 8

Answer

To find the remainder when \( 73 \times 75 \times 78 \times 57 \times 197 \) is divided by 34, we can follow these steps:

### Step 1: Find the remainders of each number when divided by 34.

1. **Calculate \( 73 \mod 34 \)**:
\[ 73 \div 34 = 2 \quad \text{(quotient)} \]
\[ 73 - (34 \times 2) = 73 - 68 = 5 \quad \Rightarrow \quad 73 \mod 34 = 5 \]

2. **Calculate \( 75 \mod 34 \)**:
\[ 75 \div 34 = 2 \quad \text{(quotient)} \]
\[ 75 - (34 \times 2) = 75 - 68 = 7 \quad \Rightarrow \quad 75 \mod 34 = 7 \]

3. **Calculate \( 78 \mod 34 \)**:
\[ 78 \div 34 = 2 \quad \text{(quotient)} \]
\[ 78 - (34 \times 2) = 78 - 68 = 10 \quad \Rightarrow \quad 78 \mod 34 = 10 \]

4. **Calculate \( 57 \mod 34 \)**:
\[ 57 \div 34 = 1 \quad \text{(quotient)} \]
\[ 57 - (34 \times 1) = 57 - 34 = 23 \quad \Rightarrow \quad 57 \mod 34 = 23 \]

5. **Calculate \( 197 \mod 34 \)**:
\[ 197 \div 34 = 5 \quad \text{(quotient)} \]
\[ 197 - (34 \times 5) = 197 - 170 = 27 \quad \Rightarrow \quad 197 \mod 34 = 27 \]

### Step 2: Substitute the remainders into the product.

Now we can substitute these remainders back into the product:
\[ (73 \times 75 \times 78 \times 57 \times 197) \mod 34 = (5 \times 7 \times 10 \times 23 \times 27) \mod 34 \]

### Step 3: Calculate the product step by step.

1. **Calculate \( 5 \times 7 \)**:
\[ 5 \times 7 = 35 \]
\[ 35 \mod 34 = 1 \]

2. **Multiply by 10**:
\[ 1 \times 10 = 10 \]

3. **Multiply by 23**:
\[ 10 \times 23 = 230 \]
\[ 230 \mod 34 = 28 \quad \text{(since } 230 - (34 \times 6) = 230 - 204 = 26\text{)} \]

4. **Multiply by 27**:
\[ 28 \times 27 = 756 \]
\[ 756 \mod 34 = 14 \quad \text{(since } 756 - (34 \times 22) = 756 - 748 = 8\text{)} \]

### Final Step: Find the remainder.
Thus, the remainder when \( 73 \times 75 \times 78 \times 57 \times 197 \) is divided by 34 is \( 8 \).