What are examples of relatively prime numbers

Prime numbers: explanation, examples and calculation

Prime numbers are covered here. Let's look at this:

  • Explanationswhat a prime number is and how to compute a prime number.
  • Lots Examples to prime numbers.
  • tasks / Exercises to this topic.
  • A Video to prime numbers.
  • A Question and answer area to this area.

We'll look at the prime numbers in a moment. We also take a look at how you can check for yourself whether a number is prime or not. To do this you need the rules of divisibility. With these you can find out whether a number is divisible by another number or a remainder. If you have no idea about it, read it briefly.

Explanation of prime numbers

Let's start with an explanation of prime numbers. First of all, everyone should understand what that is. A definition for a prime number:

Note:
  • A prime number is a number that is only divisible by itself and by 1 with no remainder.
  • A prime number is always a natural number.
  • However, the 0 and 1 are not prime numbers.

The first prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53.

Why are 0 and 1 not prime?

Let's start with the question, why is 0 not a prime number? This is relatively easy because a number has to be divisible by itself. This is not the case with zero, since one cannot divide by zero. The calculation of the task 0: 0 is not allowed.

And why is 1 not a prime number? Well, there were times in mathematics when 1 was considered a prime number. Because the 1 can be divided by 1 and by itself without a remainder. These criteria are therefore met. Nevertheless, in the course of the last century, by definition, it was decided to no longer regard 1 as a prime number. The reason for this was, for example, that 1 has only one factor while the other prime numbers have two factors. In addition, the prime factorization with a 1 would not be unique (brief information on this below).

How do you check whether a number is prime?

How can you find out if a number is prime or not? There are various ways of doing a prime number test. We would like to briefly introduce a simple procedure here. The method:

  1. You take the number to be examined and extract the root from it.
  2. You write down which prime numbers there are up to this number
  3. Apply these prime numbers to the original number and see if they divide without a remainder.
  4. If no number is found by then, it is a prime number.

example 1:

Is the number 163 a prime number?

Solution:

We first take the root of the number 163. This is approximately 12,767.

We look for all prime numbers up to this number (look at the beginning of the article). These are 2, 3, 5, 7 and 11. We now take the 163 and divide by all these prime numbers. If there is no remainder (is identical to zero after the decimal point) we have a divisor. If there is a remainder (we have something behind the decimal point not equal to zero) the prime number is not a divisor. Let's do the math:

As you can clearly see: We always have numbers after the decimal point. There is no division with only zeros after the decimal point. As this is at all Calculations was the case 163 is a prime number.

Example 2:

Is the number 228 a prime number?

Solution:

We take the root of the number 228 and get something like 15.1.

Up to this number there are the prime numbers 2, 3, 5, 7, 11 and 13. So we take the 228 and divide it by these prime numbers. If there is no remainder somewhere, we do not have a prime number.

As you can see, we have two divisions with no remainder (framed in green). This is why 228 not a prime number.

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Prime numbers examples / lists

There are numerous examples of lists / tables of prime numbers in this section. These lists are interesting because some people look directly for lists from prime numbers up to 50, 100 or even 1000.

Prime numbers up to 50:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Prime numbers up to 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Prime numbers up to 1000:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997

Application of prime numbers:

Where are prime numbers actually used? Here are a few examples:

  • Prime factorization: The prime factors or prime factorization are about dividing a number into the smallest possible multiplications of prime numbers. The number 90 can be broken down into 90 = 2 · 5 · 3 · 3. Where 2, 5 and 3 are each prime numbers.
  • gcd: The gcd stands for greater common factor. Two numbers are broken down and the largest possible common number is then sought. Take the numbers 36 and 48 as an example. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36 and the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The number 12 is the largest number that occurs in both factors.
  • LCM: The LCM stands for the smallest common multiple. Here, too, two numbers are examined. Each number is multiplied by 2, 3, 4 etc. and written down. This is followed by a look to find out where the lowest common number can be found. The LCM of 12 and 18 is to be determined as an example. The multiples of 12 are 12, 24, 36, 48, 60 .... The multiples of 18 are 18, 36, 54, 72, 90. The smallest common number is therefore 36.

Tasks / exercises prime numbers

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Video prime numbers

Explanations of prime numbers

This video is about.

  • What is a prime number anyway.
  • Examples of prime numbers.
  • Finding out if a number is prime.

Feel free to recalculate the examples from the video yourself.


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Questions and answers prime number

This section deals with typical prime number questions and answers.

Q: Is there a greatest prime number?

A: No there isn't. According to Euclid's theorem, there is no greatest prime number. One can thus - with computers - always find even larger prime numbers.

Q: What are the primality test procedures?

There are numerous procedures and background articles that deal with prime numbers, related topics, and background knowledge. You are welcome to take a look at the following areas for the prime number test.

  • Trial division
  • Sieve of Eratosthenes
  • Atkin sieve
  • Fermatscher primality test
  • Miller-Rabin test