# Prime numbers: Factorisation

Prime numbers have practical applications every day because...

## Prime numbers: Factorisation

There's something special about prime numbers. These numbers can be divided only by 1 and themselves. You haven't heard of prime numbers? Watch the video on rectangular numbers first and then come back here. This is Euclid, one of the ancient Greeks.

He lived in Alexandria in modern Egypt about 2,300 years ago. Even then, mathematicians knew there was something strange about the numbers you can't make rectangles with. Into the subject of prime numbers still poses several unresolved riddles, which keep mathematicians up at night. This is because it's not easy to tell if a very large number is prime. Because it takes a lot of time to tell which numbers are prime, cryptography uses this fact to secure data transfers over the Internet.

These are all prime numbers less than 100. They don't appear to come regularly, do they? But there are actually several patterns among prime numbers. All prime numbers except the first one are odd numbers, and they have to be, because a prime number is an integer that can be divided only by 1 and itself. And even numbers can be divided by 2, so even numbers except 2 can't be prime.

The next pattern is that prime numbers often come in pairs. In other words, two consecutive odd numbers sometimes are both prime. Then they are called twin primes. This is not very helpful but remarkable property has been puzzling mathematicians for thousands of years. The third pattern is that prime numbers are more frequent in the beginning, but then the distance between them gets larger.

But even if there's more space between prime numbers as they get larger, they never end. Euclid proved that there's an infinite number of prime numbers. Numbers that are not prime, those that you can make rectangles with are called composite numbers. You can evenly divide such numbers by themselves and 1. something other than Take a composite number - take 18.

It can be resolved into factors, factorized. 2 x 9 is 18 and a rectangle. 2 is a prime number, so you can't factorize it, but 9 can be factorized as 3 x 3, a nice square. Now only prime numbers are left. You factorized 18 into prime numbers, 2 x 3 x 3 If you turn the picture upside down, it looks like a tree.

We call it a factor tree. Even numbers can be immediately divided by 2 and if you remember the number from the times tables, you can factorize it quickly. 98 is even. Its 2 x 49 and as you know from the 7-times table, 7 x 7 is 49.