Kummer's congruences
Kummer's congruences are like a special kind of puzzle for numbers. They help us find answers when we divide bigger numbers by smaller numbers and we want to know what the remainder is. Let me explain how it works.
Imagine you have a bunch of apples, let's say 18 apples, and you want to share them equally among your friends. But you have 5 friends, and 18 is not exactly divisible by 5. So how do you solve this problem and figure out how many apples each friend will get?
This is where Kummer's congruences come in. Kummer discovered a way to solve this problem using a special equation that helps us find the remainder when we divide. Let me break it down for you.
First, we write down the number of apples we have, which is 18, and we write down the number of friends we want to share them with, which is 5.
Then, we take the bigger number, which is 18, and divide it by the smaller number, which is 5. We get a quotient and a remainder. The quotient is the number of apples each friend will get, and the remainder is the number of apples left over.
In our example, when we divide 18 by 5, we get a quotient of 3 and a remainder of 3. This means that each friend will get 3 apples, and there will be 3 apples left over.
But how does Kummer's congruences help us find this quickly? Well, Kummer figured out that we can use some special formulas to find the remainder without doing a long division.
For example, Kummer's first congruence says that if we have a number n and we want to divide it by a prime number p, the remainder we get will be the same as the remainder we would get if we divide the digits of n by p and add them up.
Let's see how this works with our example. We have the number 18, and we want to divide it by the prime number 5. So, we take the digits of 18, which are 1 and 8, and we add them up. 1 + 8 = 9. Now we divide 9 by 5, and we get a remainder of 4. And guess what? This remainder is the same as the remainder we got earlier when we did the long division!
Kummer's other congruences work in a similar way. They help us find the remainder when we divide by a prime number without doing the long division.
So, in summary, Kummer's congruences are special formulas that help us find the remainder when we divide bigger numbers by smaller numbers without actually doing the division. They are like puzzle-solving equations that make it easier for us to share things equally among our friends!
Imagine you have a bunch of apples, let's say 18 apples, and you want to share them equally among your friends. But you have 5 friends, and 18 is not exactly divisible by 5. So how do you solve this problem and figure out how many apples each friend will get?
This is where Kummer's congruences come in. Kummer discovered a way to solve this problem using a special equation that helps us find the remainder when we divide. Let me break it down for you.
First, we write down the number of apples we have, which is 18, and we write down the number of friends we want to share them with, which is 5.
Then, we take the bigger number, which is 18, and divide it by the smaller number, which is 5. We get a quotient and a remainder. The quotient is the number of apples each friend will get, and the remainder is the number of apples left over.
In our example, when we divide 18 by 5, we get a quotient of 3 and a remainder of 3. This means that each friend will get 3 apples, and there will be 3 apples left over.
But how does Kummer's congruences help us find this quickly? Well, Kummer figured out that we can use some special formulas to find the remainder without doing a long division.
For example, Kummer's first congruence says that if we have a number n and we want to divide it by a prime number p, the remainder we get will be the same as the remainder we would get if we divide the digits of n by p and add them up.
Let's see how this works with our example. We have the number 18, and we want to divide it by the prime number 5. So, we take the digits of 18, which are 1 and 8, and we add them up. 1 + 8 = 9. Now we divide 9 by 5, and we get a remainder of 4. And guess what? This remainder is the same as the remainder we got earlier when we did the long division!
Kummer's other congruences work in a similar way. They help us find the remainder when we divide by a prime number without doing the long division.
So, in summary, Kummer's congruences are special formulas that help us find the remainder when we divide bigger numbers by smaller numbers without actually doing the division. They are like puzzle-solving equations that make it easier for us to share things equally among our friends!