As you may already know, a prime number is one that has exactly two factors, 1 and itself. No number will divide a prime number and create a non decimal answer. It turns out too, that when you break down a number into its factors, the factors are always prime, whether the number itself is or not. This is especially useful when doing square roots. For example, when you factor the number 36, you get 2*2*3*3. This also means the root of 36 is root(2*2*3*3). Since the square root of a number times itself is just the number, it is easy to see that for each repeated number under the square root sign, we can pull one out as a whole number. So the square root of 36 is 2*3, or 6.
The same thing can be done for prime numbers, but you won't get any actual factors, and it would be silly to write 1 as a factor; it's kind of a given. Nonetheless, this same thing can be done for much larger numbers too. If we wanted to find root(220,000) we would of course need a calculator and the best place to start would be dividing by 2 because it is obvious that 2 will divide it. Anyway, we find that the factors are 2*2*2*2*2*5*5*5*5*11. So root(220,000) equals 2*2*5*5*root(5*11) or 100*root(55).
Now get ready for the really cool part. You probably already realize that you can do this same thing, at least in theory, for incredibly huge numbers with as many digits as you want. Even a number with a trillion digits could be factored in this way. But what about an incredibly huge number that just happens to be prime? It has no factors, but we it would take a lot of effort that could only be accomplished by a computer to even figure that out. Computers also have their limits, so if we get a big enough number, like one with 100's of millions of digits, then even a computer isn't fast enough to ever come up with an answer.
Governments, phone companies, and other organizations with sensitive information us prime numbers in a way that codes their information in an almost unbreakable way. They take two huge prime numbers and multiply them together, effectively creating a number whose only factors are those two huge primes. Anyone can know this number because only by knowing the factors is the information retrievable. If the correct recipient is the only one who knows one of the factors, the other is easily attainable, and the code can be read.
The same thing can be done for prime numbers, but you won't get any actual factors, and it would be silly to write 1 as a factor; it's kind of a given. Nonetheless, this same thing can be done for much larger numbers too. If we wanted to find root(220,000) we would of course need a calculator and the best place to start would be dividing by 2 because it is obvious that 2 will divide it. Anyway, we find that the factors are 2*2*2*2*2*5*5*5*5*11. So root(220,000) equals 2*2*5*5*root(5*11) or 100*root(55).
Now get ready for the really cool part. You probably already realize that you can do this same thing, at least in theory, for incredibly huge numbers with as many digits as you want. Even a number with a trillion digits could be factored in this way. But what about an incredibly huge number that just happens to be prime? It has no factors, but we it would take a lot of effort that could only be accomplished by a computer to even figure that out. Computers also have their limits, so if we get a big enough number, like one with 100's of millions of digits, then even a computer isn't fast enough to ever come up with an answer.
Governments, phone companies, and other organizations with sensitive information us prime numbers in a way that codes their information in an almost unbreakable way. They take two huge prime numbers and multiply them together, effectively creating a number whose only factors are those two huge primes. Anyone can know this number because only by knowing the factors is the information retrievable. If the correct recipient is the only one who knows one of the factors, the other is easily attainable, and the code can be read.
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