How to Find if a Number is Prime or Composite?
To determine whether a given number is prime or composite, you can follow these methods:
Trial Division Method
- Divide the number n by all integers from 2 to n.
- If any of these divisions result in a whole number, then n is composite.
- If none of the divisions result in a whole number, then n is prime.
Example: Let’s determine if n=17 is prime or composite:
- We divide n by integers from 2 to 17, which is approximately 4.12.
- We check divisions by 2, 3, and 4. Since none of these divisions result in a whole number, n=17 is prime.
Sieve of Eratosthenes
- Create a list of integers from 2 to the number n.
- Start with the first number in the list (2) and mark all of its multiples as composite.
- Move to the next unmarked number in the list and repeat the process until you reach n.
- If the number is unmarked after the process, it is prime; otherwise, it is composite.
Example: Let’s determine if n=29 is prime or composite using the Sieve of Eratosthenes:
- We create a list of integers from 2 to 29.
- We start with 2 and mark all its multiples (4, 6, 8, …).
- We continue with 3 (already marked), then 5, 7, 11, 13, 17, 19, 23, and finally 29.
- Since 29 is unmarked, it is prime.
Fermat’s Little Theorem
This method is usually used for large numbers, especially in cryptography.
- If an−1 mod n=1 for some integer a, then n is likely prime.
- If an−1 mod n ≠ 1, then n is composite.
- This method is probabilistic and not conclusive but is used in certain algorithms.
These methods provide efficient ways to determine whether a given number is prime or composite, depending on the size of the number and the available computational resources.
Prime and Composite Numbers
Prime and Composite Numbers are commonly used classifications of Natural Numbers based on divisibility and the number of Factors. A Prime Number has only two factors while Composite Numbers have more than two factors. This classification of Numbers makes the study of natural numbers more organized and convenient and is useful in a variety of situations like computer algorithms, biology, understanding of Number Theory, etc.
This article describes what are prime and composite numbers, the types of primes and composite numbers, and tests to check whether a given number is prime or not (primality tests). Finally, a few solved questions, and a few practice problems related to prime and composite numbers are presented.
Table of Content
- What are Prime and Composite Numbers?
- Types of Prime and Composite numbers
- Prime and Composite Numbers from 1 to 100
- Prime and Composite Numbers chart
- Difference between Prime and Composite Numbers
- Tests for Prime and Composite numbers
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