What remainder does the product 2005·2006·2007·2008·2009·2010 leave after division by 11?

To find the remainder when the product \( 2005 \cdot 2006 \cdot 2007 \cdot 2008 \cdot 2009 \cdot 2010 \) is divided by 11, we will apply **Fermat's Little Theorem** and properties of congruences.

### Step 1: Find each term modulo 11
First, calculate each number modulo 11:
\[
2005 \equiv 3 \pmod{11} \quad \text{(since \( 2005 - 3 = 2002 \) is divisible by 11)}
\]
\[
2006 \equiv 4 \pmod{11} \quad \text{(since \( 2006 - 4 = 2002 \))}
\]
\[
2007 \equiv 5 \pmod{11} \quad \text{(since \( 2007 - 5 = 2002 \))}
\]
\[
2008 \equiv 6 \pmod{11} \quad \text{(since \( 2008 - 6 = 2002 \))}
\]
\[
2009 \equiv 7 \pmod{11} \quad \text{(since \( 2009 - 7 = 2002 \))}
\]
\[
2010 \equiv 8 \pmod{11} \quad \text{(since \( 2010 - 8 = 2002 \))}
\]

### Step 2: Calculate the product modulo 11
We now need to find the product:
\[
3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \pmod{11}
\]
We'll multiply step-by-step:
\[
3 \cdot 4 = 12 \equiv 1 \pmod{11}
\]
\[
1 \cdot 5 = 5 \pmod{11}
\]
\[
5 \cdot 6 = 30 \equiv 8 \pmod{11}
\]
\[
8 \cdot 7 = 56 \equiv 1 \pmod{11}
\]
\[
1 \cdot 8 = 8 \pmod{11}
\]

### Final Answer
The remainder when \( 2005 \cdot 2006 \cdot 2007 \cdot 2008 \cdot 2009 \cdot 2010 \) is divided by 11 is:
\[
\boxed{8}
\]

Would you like more details or have any questions?

### Related Questions
1. What is Fermat's Little Theorem, and how is it applied in modular arithmetic problems?
2. How can we generalize finding remainders of large products modulo smaller numbers?
3. Can this method be used for any prime modulus, or are there exceptions?
4. How does modular arithmetic simplify calculations in number theory?
5. Why is understanding modular arithmetic crucial in cryptography?

### Tip
When dealing with products and sums modulo a number, breaking down complex expressions step-by-step simplifies calculations significantly.

Find the remainder when the product of 2005·2006·2007·2008·2009·2010 is divided by 11. This solution uses modular arithmetic and properties of congruences, applying Fermat's Little Theorem to simplify calculations. Suitable for high school students learning number theory.

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