Find the remainder when `1719 xx 1721 xx 1723 xx 1725 xx 1727` is divided by 18.

Answer

To find the remainder when \(1719 \times 1721 \times 1723 \times 1725 \times 1727\) is divided by 18, we can follow these steps:

### Step 1: Find the remainders of each number when divided by 18.
1. **Calculate \(1719 \mod 18\)**:
\[ 1719 \div 18 = 95 \quad \text{(quotient)} \]
\[ 1719 - (18 \times 95) = 1719 - 1710 = 9 \quad \text{(remainder)} \]
So, \(1719 \equiv 9 \mod 18\).

2. **Calculate \(1721 \mod 18\)**:
\[ 1721 \div 18 = 95 \quad \text{(quotient)} \]
\[ 1721 - (18 \times 95) = 1721 - 1710 = 11 \quad \text{(remainder)} \]
So, \(1721 \equiv 11 \mod 18\).

3. **Calculate \(1723 \mod 18\)**:
\[ 1723 \div 18 = 95 \quad \text{(quotient)} \]
\[ 1723 - (18 \times 95) = 1723 - 1710 = 13 \quad \text{(remainder)} \]
So, \(1723 \equiv 13 \mod 18\).

4. **Calculate \(1725 \mod 18\)**:
\[ 1725 \div 18 = 95 \quad \text{(quotient)} \]
\[ 1725 - (18 \times 95) = 1725 - 1710 = 15 \quad \text{(remainder)} \]
So, \(1725 \equiv 15 \mod 18\).

5. **Calculate \(1727 \mod 18\)**:
\[ 1727 \div 18 = 95 \quad \text{(quotient)} \]
\[ 1727 - (18 \times 95) = 1727 - 1710 = 17 \quad \text{(remainder)} \]
So, \(1727 \equiv 17 \mod 18\).

### Step 2: Multiply the remainders.
Now we will multiply the remainders we found:
\[ (9 \times 11 \times 13 \times 15 \times 17) \mod 18 \]

### Step 3: Calculate the product step by step.
1. **Calculate \(9 \times 11\)**:
\[ 9 \times 11 = 99 \]
\[ 99 \mod 18 = 9 \quad \text{(since } 99 - 18 \times 5 = 9\text{)} \]

2. **Multiply by \(13\)**:
\[ 9 \times 13 = 117 \]
\[ 117 \mod 18 = 3 \quad \text{(since } 117 - 18 \times 6 = 3\text{)} \]

3. **Multiply by \(15\)**:
\[ 3 \times 15 = 45 \]
\[ 45 \mod 18 = 9 \quad \text{(since } 45 - 18 \times 2 = 9\text{)} \]

4. **Multiply by \(17\)**:
\[ 9 \times 17 = 153 \]
\[ 153 \mod 18 = 9 \quad \text{(since } 153 - 18 \times 8 = 9\text{)} \]

### Step 4: Final remainder.
The final product modulo 18 is:
\[ 153 \mod 18 = 9 \]

### Conclusion:
The remainder when \(1719 \times 1721 \times 1723 \times 1725 \times 1727\) is divided by 18 is **9**.

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