What is the smallest natural number n such that the number N=100000x100002x100006x100008+n is a perfect square?

We are tasked with finding the smallest natural number \( n \) such that the number 

\[
N = 100000 \times 100002 \times 100006 \times 100008 + n
\]

is a perfect square.

### Step 1: Simplifying the expression for \( N \)
First, we note that the four numbers in the product are quite close to each other. To simplify the problem, we approximate them around a central value. The average of these numbers is \( 100004 \), so we express the numbers in terms of this average:

\[
100000 = 100004 - 4, \quad 100002 = 100004 - 2, \quad 100006 = 100004 + 2, \quad 100008 = 100004 + 4.
\]

This allows us to write the product as:

\[
N = (100004 - 4)(100004 - 2)(100004 + 2)(100004 + 4).
\]

### Step 2: Using difference of squares
We can apply the difference of squares formula:

\[
(a - b)(a + b) = a^2 - b^2.
\]

Applying this to our expression, we first pair the terms:

\[
(100004 - 4)(100004 + 4) = 100004^2 - 4^2 = 100004^2 - 16,
\]
\[
(100004 - 2)(100004 + 2) = 100004^2 - 2^2 = 100004^2 - 4.
\]

Thus, the product becomes:

\[
N = (100004^2 - 16)(100004^2 - 4).
\]

### Step 3: Expanding the product
Now we expand the product of these two terms:

\[
N = (100004^2 - 16)(100004^2 - 4) = 100004^4 - 4 \times 100004^2 - 16 \times 100004^2 + 16 \times 4.
\]

Simplifying further:

\[
N = 100004^4 - 20 \times 100004^2 + 64.
\]

Thus, we have:

\[
N = 100004^4 - 20 \times 100004^2 + 64 + n.
\]

### Step 4: Making \( N \) a perfect square
For \( N \) to be a perfect square, we want the expression \( 100004^4 - 20 \times 100004^2 + 64 + n \) to equal some perfect square \( k^2 \). To solve this, we can try specific values of \( n \) such that the whole expression becomes a perfect square.

Let me calculate and find the smallest \( n \) that satisfies this condition.The smallest natural number \( n \) such that 

\[
N = 100000 \times 100002 \times 100006 \times 100008 + n
\]

is a perfect square is \( n = 36 \).

Would you like more details on the steps, or do you have any other questions?

Here are five related questions to expand on this topic:
1. How do you determine if a number is a perfect square?
2. What is the difference of squares, and how is it applied in problems like this?
3. How can you approximate large numbers using central values?
4. Why is the product of numbers around a central value useful in simplification?
5. What are some other methods to solve similar number theory problems?

**Tip:** When dealing with large numbers, using approximations or algebraic identities like the difference of squares can significantly simplify calculations.

Discover how to find the smallest natural number n that makes N = 100000 x 100002 x 100006 x 100008 + n a perfect square. This solution applies the difference of squares formula and expands products, suitable for students in Grades 11-12.

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