# Relatively prime.

mathematics textbooks sometimes difficult to understand.Dry and clear language authors are not always easy to understand.And there are always interrelated topics, vzaimovytekayuschie.To develop a single topic have to raise a number of previous and sometimes flip through the whole textbook.Complicated?Yes.Let's dare to circumvent these difficulties and try to find a topic not quite the standard approach.We make a kind of excursion into the country numbers.The definition, however, we still remain the same, because the rules of mathematics can not be undone.Thus, relatively prime numbers - natural numbers with common divisor equal to one.It's clear?It is.

For a good example, let's take the number 6 and 13. And then, and more - are divisible by one (relatively prime).But the numbers 12 and 14 - can not be established, as divided not only to 1 but also to 2. The following numbers - 21 and 47 are also not suitable for the category of "relatively prime": they can be divided not only one, buteven at 7.

Indicate relatively prime because: ( and , y) = 1.

We can say even simpler: the common divisor (the highest) is equal to one.
What are we learning?Reasons enough.

mutually prime numbers included in some encryption system.Those who work with the Hill cipher, or the system of substitutions Caesar, understand that without this knowledge - anywhere.If you've heard of the random number generator is unlikely to dare to deny: relatively prime numbers are used and there.

Now let's talk about how to obtain these numbers.The numbers are simple, as you know, can have only two divisors: they divide by themselves and by one.Say, 11, 7, 5, 3 - the number of simple, but 9 - no, it's already number divisible and 9, and 3, and 1.

And if and - a prime number, and have - from the set {1, 2, ... and - 1}, then guaranteed ( and , have ) = 1, or relatively prime - and and have .

It is, rather, not even an explanation and repetition or summarizing what has been said.

Getting primes sieve of Eratosthenes is possible, however, for the impressive numbers (billions, for example), this method is too long, but, unlike the super-formula, which sometimes make mistakes, more reliable.

can work by selecting from & gt; and .To do this chosen so that the number on the and not divided.For this number is simply multiplied by the number of natural and added (or, on the contrary, is deducted) the amount (say, p ), which is less than and :

y = p a + k

If, for example, and = 71, p = 3, q ​​= 10, then, accordingly, have here is equal to 713. There is another choice, with degrees.

composite number, in contrast to the relatively prime, and divided themselves, and at 1, and the other numbers (also without a trace).

In other words, the natural numbers (except one) divided into components and simple.

Primes - the number of natural, non-trivial (distinct from the numbers and units) dividers.Especially important is their role in today's modern, fast-paced cryptography, number theory by which, previously thought very abstract discipline, has become so in demand: data protection algorithms are constantly being improved.

largest prime number found ophthalmologist Dr. Martin Nowak, who participated in the project GIMPS (distribution calculation), together with other enthusiasts, who numbered about 15 thousand. In the calculations took six years.It involved two dozen computers in the eye clinic Novak.The result of titanic work and perseverance was the number 225964951-1, writing it in a 7,816,230-decimal places.By the way, the record for the large number was delivered six months before this discovery.And there were signs on the lower half.

have genius who wants to call a number where length of decimal notation, "jump" ten-mark, there is a chance to get not only international fame but also \$ 100 000.By the way, the numbers overcame millionth milestone marks Nayan Hayratval received a lower amount (50 000 dollars).

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