What is the remainder when `1719 xx 1715 xx 1713` is divided by 17 ?

Options

A. 1
B. 16
C. 3
D. none of these

Answer

To find the remainder when \( 1719 \times 1715 \times 1713 \) is divided by \( 17 \), we can use the property of modular arithmetic. Instead of calculating the entire product, we can find the remainders of each number when divided by \( 17 \) and then multiply those remainders together.

### Step-by-Step Solution:

1. **Calculate the remainder of 1719 when divided by 17**:
\[ 1719 \div 17 = 101 \quad \text{(the quotient)} \]
\[ 101 \times 17 = 1717 \quad \text{(the largest multiple of 17 less than 1719)} \]
\[ 1719 - 1717 = 2 \]
Thus, \( 1719 \mod 17 = 2 \).

2. **Calculate the remainder of 1715 when divided by 17**:
\[ 1715 \div 17 = 101 \]
\[ 101 \times 17 = 1717 \]
\[ 1715 - 1717 = -2 \quad \text{(since we are looking for a positive remainder, we add 17)} \]
\[ -2 + 17 = 15 \]
Thus, \( 1715 \mod 17 = 15 \).

3. **Calculate the remainder of 1713 when divided by 17**:
\[ 1713 \div 17 = 101 \]
\[ 101 \times 17 = 1717 \]
\[ 1713 - 1717 = -4 \quad \text{(again, we add 17)} \]
\[ -4 + 17 = 13 \]
Thus, \( 1713 \mod 17 = 13 \).

4. **Multiply the remainders**:
Now we multiply the remainders we found:
\[ (1719 \mod 17) \times (1715 \mod 17) \times (1713 \mod 17) = 2 \times 15 \times 13 \]

5. **Calculate \( 2 \times 15 \)**:
\[ 2 \times 15 = 30 \]

6. **Calculate \( 30 \times 13 \)**:
\[ 30 \times 13 = 390 \]

7. **Find the remainder of 390 when divided by 17**:
\[ 390 \div 17 = 22 \quad \text{(the quotient)} \]
\[ 22 \times 17 = 374 \]
\[ 390 - 374 = 16 \]
Thus, the remainder when \( 1719 \times 1715 \times 1713 \) is divided by \( 17 \) is \( 16 \).

### Final Answer:
The remainder when \( 1719 \times 1715 \times 1713 \) is divided by \( 17 \) is \( \boxed{16} \).