What is the remainder when `1719xx1715xx1713` 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 which states that \( (a \times b) \mod m = [(a \mod m) \times (b \mod m)] \mod m \).

### Step 1: Calculate \( 1719 \mod 17 \)

First, we need to find the remainder of \( 1719 \) when divided by \( 17 \).

\[ 1719 \div 17 \approx 101.1176 \quad \text{(take the integer part, which is 101)} \]
\[ 101 \times 17 = 1717 \]
\[ 1719 - 1717 = 2 \]

So,

\[ 1719 \mod 17 = 2 \]

### Step 2: Calculate \( 1715 \mod 17 \)

Next, we find the remainder of \( 1715 \) when divided by \( 17 \).

\[ 1715 \div 17 \approx 101.3529 \quad \text{(take the integer part, which is 101)} \]
\[ 101 \times 17 = 1717 \]
\[ 1715 - 1717 = -2 \quad \text{(to make it positive, we can add 17)} \]
\[ -2 + 17 = 15 \]

So,

\[ 1715 \mod 17 = 15 \]

### Step 3: Calculate \( 1713 \mod 17 \)

Now, we find the remainder of \( 1713 \) when divided by \( 17 \).

\[ 1713 \div 17 \approx 100.7647 \quad \text{(take the integer part, which is 100)} \]
\[ 100 \times 17 = 1700 \]
\[ 1713 - 1700 = 13 \]

So,

\[ 1713 \mod 17 = 13 \]

### Step 4: Combine the results

Now we can combine the results using the modular arithmetic property:

\[ 1719 \times 1715 \times 1713 \mod 17 = (2 \times 15 \times 13) \mod 17 \]

### Step 5: Calculate \( 2 \times 15 \times 13 \)

First, calculate \( 2 \times 15 \):

\[ 2 \times 15 = 30 \]

Now multiply by \( 13 \):

\[ 30 \times 13 = 390 \]

### Step 6: Find \( 390 \mod 17 \)

Now we need to find the remainder of \( 390 \) when divided by \( 17 \):

\[ 390 \div 17 \approx 22.9412 \quad \text{(take the integer part, which is 22)} \]
\[ 22 \times 17 = 374 \]
\[ 390 - 374 = 16 \]

So,

\[ 390 \mod 17 = 16 \]

### Final Answer

The remainder when \( 1719 \times 1715 \times 1713 \) is divided by \( 17 \) is \( 16 \).

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