Multiplicative group of integers modulo n

Okay kiddo! So, let's say you are playing with some toys and you have a bunch of them. Now, if we group them in some way where we only count some of them and then we can find a way to make them "play together" in a special way, that would be like the multiplicative group of integers modulo n!

Now, let's talk about the numbers. When we say "integers modulo n", what we mean is that we are only looking at a certain group of numbers that all have something in common. Think of it like a special club where only certain numbers are allowed in.

For example, let's say we only want to look at numbers that are less than 6. So, our club would have the numbers 0, 1, 2, 3, 4, and 5. We call them "integers modulo 6". Now, we want to find a way to make them play together.

The special thing about this club is that we can't just add or subtract like we normally do with numbers. Instead, we have to use a special rule that involves multiplying and dividing. We call this rule "modulo" or "mod" for short.

So, let's say we want to find the multiplicative group of integers modulo 6. We start by listing all the numbers in our club: 0, 1, 2, 3, 4, and 5. Now, we look for numbers that "play nicely" when we use the mod rule.

For example, 1 is easy, because when we multiply it by any number and then apply mod 6, we get the same number back. So, 1 is definitely in our group. What about 2? Well, if we multiply 2 by any even number, we get an even number and if we multiply it by any odd number, we get an odd number. So, 2 is definitely not in our group.

We keep going like this, checking all the numbers and seeing which ones "play nicely" with the mod rule. In the end, we get a group of numbers that we can "play" with using multiplication and modular arithmetic. This is our multiplicative group of integers modulo n!

And that's it! We now have our special group of "playful" numbers that we can use to do some math.