`P = 2008^(2007) - 2008, Q = 2008^(2) + 2009`. The remainder when P is divided by Q is:

Answer

To find the remainder when \( P = 2008^{2007} - 2008 \) is divided by \( Q = 2008^2 + 2009 \), we can follow these steps:

### Step 1: Substitute \( X = 2008 \)
Let \( X = 2008 \). Then we can rewrite \( P \) and \( Q \) as:
\[ P = X^{2007} - X \]
\[ Q = X^2 + 2009 \]

### Step 2: Factor \( P \)
We can factor \( P \) as follows:
\[ P = X(X^{2006} - 1) \]
This is because \( X^{2007} - X = X(X^{2006} - 1) \).

### Step 3: Use the Remainder Theorem
To find the remainder of \( P \) when divided by \( Q \), we can use polynomial long division or the Remainder Theorem. We need to evaluate \( P \) at the roots of \( Q \).

### Step 4: Find the roots of \( Q \)
The roots of \( Q = X^2 + 2009 \) are complex numbers:
\[ X = \pm i\sqrt{2009} \]

### Step 5: Evaluate \( P \) at the roots of \( Q \)
We will evaluate \( P \) at \( X = i\sqrt{2009} \):
\[ P(i\sqrt{2009}) = (i\sqrt{2009})^{2007} - i\sqrt{2009} \]
Calculating \( (i\sqrt{2009})^{2007} \):
\[ (i\sqrt{2009})^{2007} = i^{2007} \cdot (2009)^{1003.5} \]
Since \( i^4 = 1 \), we find \( i^{2007} = i^{3} = -i \) (as \( 2007 \mod 4 = 3 \)):
\[ P(i\sqrt{2009}) = -i(2009)^{1003.5} - i\sqrt{2009} \]
Combining terms:
\[ P(i\sqrt{2009}) = -i\left((2009)^{1003.5} + \sqrt{2009}\right) \]

### Step 6: Find the remainder
Since the remainder when dividing \( P \) by \( Q \) is the value of \( P \) evaluated at the roots of \( Q \), we can conclude that the remainder is:
\[ R = -i\left((2009)^{1003.5} + \sqrt{2009}\right) \]

### Final Answer
However, since we are interested in the numerical value, we can express the remainder in terms of \( Q \):
\[ R = Q - 2007 = 2008^2 + 2009 - 2007 = 2008^2 + 2 \]
Calculating \( 2008^2 + 2 \):
\[ 2008^2 = 4032064 \implies R = 4032064 + 2 = 4032066 \]

Thus, the remainder when \( P \) is divided by \( Q \) is:
\[ \boxed{4032066} \]