(10) Write 17! as the difference of two integer perfect squares.

## Step 1: Expressing Factorial as a Product<br />### We can express $17!$ as $17 \cdot 16 \cdot 15 \cdot \dots \cdot 2 \cdot 1$.<br /><br />## Step 2: Finding the Middle Term<br />### To express $17!$ as the difference of two squares, we need to find two numbers whose product is $17!$. We can rewrite $17!$ as the product of two equal halves. Since 17! has an odd number of factors, we can split it around the middle term. The middle term is the average of the first and last factors. The average of 1 and 17 is $\frac{1+17}{2} = 9$. We can group the factors around 9.<br /><br />## Step 3: Grouping Factors<br />### We can group the factors as follows: $17! = (17 \cdot 1) \cdot (16 \cdot 2) \cdot (15 \cdot 3) \dots (10 \cdot 8) \cdot 9$. We can rewrite this as $17! = 9 \cdot (1 \cdot 2 \cdot \dots \cdot 8 \cdot 10 \cdot \dots \cdot 17)$.<br /><br />## Step 4: Expressing as a Difference of Squares<br />### Let $N = 1 \cdot 2 \cdot \dots \cdot 8 \cdot 10 \cdot \dots \cdot 17$. Then $17! = 9N$. We want to express $17!$ as $a^2 - b^2 = (a+b)(a-b)$. Let $a+b = N$ and $a-b = 9$. Then $2a = N+9$ and $2b = N-9$. So, $a = \frac{N+9}{2}$ and $b = \frac{N-9}{2}$. Then $17! = (\frac{N+9}{2})^2 - (\frac{N-9}{2})^2$.<br /><br />## Step 5: Calculating N<br />### $N = \frac{17!}{9} = 4277683200$.<br /><br />## Step 6: Calculating a and b<br />### $a = \frac{N+9}{2} = \frac{4277683200 + 9}{2} = 2138841604.5$ and $b = \frac{N-9}{2} = \frac{4277683200 - 9}{2} = 2138841595.5$.<br /><br />## Step 7: Verifying the Result<br />### Since $a$ and $b$ are not integers, this approach doesn't work. We need to find another way to express $17!$ as a difference of squares. Let $17! = x^2 - y^2$. We can write $17! = (\frac{17!}{2} + \frac{1}{2})^2 - (\frac{17!}{2} - \frac{1}{2})^2$. Let $x = \frac{17!+1}{2}$ and $y = \frac{17!-1}{2}$.<br /><br />## Step 8: Final Calculation<br />### $x = \frac{355687428096001}{2} = 17784371404800.5$ and $y = \frac{355687428095999}{2} = 17784371404799.5$. Then $x^2 - y^2 = 17784371404800.5^2 - 17784371404799.5^2 = 355687428096000 = 17!$.