What are prime numbers? (We explain in sweets)
In the world of mathematics, there are special, “indivisible” numbers that behave like true loners. To understand how they work, let's imagine a real-life situation.
You have 7 chocolates. You want to divide them equally without offending anyone or cutting the candy with a knife. How to do this? You can give all 7 candies to one person. Or give 1 candy to seven friends. It is simply impossible to divide them equally into two, three or five - there will always be a remainder.
The number 7 is aprime number.
These are the very first prime numbers that we encounter when counting: 2, 3, 5, 7, 11, 13, 17, 19...
How do prime numbers differ from composite numbers?
If a number can be divided into something other than one and itself, it is calledcomposite. It seems to “consist” of other building blocks. Let's compare them in a table so we don't get confused.
| Number | What is it divided by without a remainder? | What number is this? |
|---|---|---|
| 5 | Only on 1 and 5. | Simple |
| 6 | At 1, at 2, at 3 and at 6. | Composite |
| 11 | Only on 1 and 11. | Simple |
| 12 | At 1, 2, 3, 4, 6 and 12. | Composite |
🔥 Test trap: Unit (1) is this a prime number?
No, one is neither a prime nor a composite number! Remember the rule: a prime number must have exactly two different divisors (one and the number itself). What is a unit divided by? Only for one. There is only one divisor. Therefore, the number 1 is a unique mathematical outcast that stands apart.
Amazing fact about deuce (2)
Look at all the even numbers: 4, 6, 8, 10, 20, 100. They are all even, which means they are at least divisible by 2. Therefore, they are all composite.
This leads to a cool rule: Two (2) is the only even prime number in the world! All other prime numbers (3, 5, 7, 11, and so on) will always be odd. If you come across the number 48937982, and you see that it ends in 2 (that is, even), you can immediately, in a second, tell that it is composite.
Sieve of Eratosthenes: how to find prime numbers?
Back in Ancient Greece, the clever mathematician Eratosthenes came up with a simple way to separate prime numbers from composite numbers. This method was called the "Sieve of Eratosthenes" because it sifts numbers like sand through a sieve.
How it works (you can repeat it on a piece of paper):
- Write down all the numbers in order from 1 to 100.
- Immediately cross out the 1 (it is neither one nor the other).
- Circle the two (2) - this is the first prime number. Then cross out all the numbers that are divisible by 2 (4, 6, 8, 10...).
- The next number not crossed out is three (3). Circle her! Now cross out all the numbers that are divisible by 3 (9, 15, 21...).
- Continue with 5 and 7.
As a result, you will only have prime numbers left uncrossed! For the first hundred there are exactly 25 of them.
Why do programmers and the Internet need prime numbers?
It seems that this is just a school theory that will never be useful in life. But this is a huge misconception!
- 🔒 Password protection: When you enter a password on a social network or pay for a purchase with a bank card on the Internet, your data is protected by cryptography (RSA algorithm).
- 🛡️ How it works: The computer takes two giant prime numbers (which can have hundreds of digits!) and multiplies them together. This turns out to be a super-huge number.
- 💻 Invulnerability: Multiplying two numbers is easy. But to do the opposite - take a huge ready-made number and guess from which two primes it was multiplied - is incredibly difficult. Even the most powerful hacker supercomputers will take thousands of years to solve such a cipher!