Euler's theorem

Hey kiddo! Have you ever heard of Euler's theorem? It's a very interesting mathematical concept, let me explain it to you like you're 5 years old.

Imagine you have a big circle, and you draw a straight line from the center of the circle to the edge. That line is called a radius. Now, if you take that radius and move it around the circle, it creates an inner area within the circle.

Euler's theorem helps us understand the relationship between the radius and the area inside the circle. It says that if you take the radius and raise it to a certain power (let's call it n), then add another number (let's call it a), and then take the remainder when you divide by another number (let's call it m), the answer will always be the same for any circle.

Sounds confusing? Don't worry, let me give you an example.

Let's say we have a circle with a radius of 3. We want to figure out what happens when we do the following:

- Raise the radius to the power of 2 (9)
- Add 5 (14)
- Divide by 7 and take the remainder (remainder of 0)

So the answer to our equation for this circle is 0.

Now let's imagine we have a different circle, with a radius of 5. We do the same equations:

- Raise the radius to the power of 2 (25)
- Add 5 (30)
- Divide by 7 and take the remainder (remainder of 2)

Even though the two circles have different radii, the answer to our equation is the same: 0.

That's Euler's theorem! It helps us understand the relationship between the radius and the area inside the circle, and that relationship stays the same no matter what size the circle is. Pretty cool, huh?