Order (number theory)

Order in number theory refers to a special property of certain numbers. Let's imagine that you have a big bag of candies, and you want to know how many times you need to reach in and grab a candy before you get the same one again. The number of times you need to grab a candy to get the same one again is the order of the bag of candies.

In math terms, the order of a number refers to how many times you need to multiply it by itself before you get to a certain result. For example, if we take the number 2 and multiply it by itself, we get 4. If we multiply 2 by itself again, we get 8. If we do it one more time, we get 16. So the order of 2 (in mod 17) is 4, because it takes 4 times of multiplying 2 by itself to get to 16 (and if we multiply 2 by itself again, we would be back at 1 again).

But what makes the order property special is that for some numbers, no matter how many times you multiply it by itself, you will never get to a certain result. For example, if we take the number 3 and multiply it by itself (mod 7), we get 2. If we do it again, we get 6. If we do it one more time, we get 4. And if we multiply 3 by itself one more time, we are back to 1 again. So the order of 3 (in mod 7) is 6.

Knowing the order of a number in mod some other number can be very useful in number theory, because it can help us solve certain equations and find patterns in sequences of numbers. It's kind of like a superpower that some numbers have!