Prime And Composite Numbers
Introduction to prime and composite numbers:
Prime Number:
A prime number is a natural number which has only 1 and the number itself as factors. For example, 2, 3, 5, 7, 11, 13, etc. are all prime numbers. Composite number is a natural number which has factors other than just 1 and the number itself. For example, 4, 6, 8, 9, 10, 12, etc. are all composite numbers. By convention, the number 1 is not prime or composite number.
Composite Number:
A number is called "composite" if it can be divided evenly into two or more parts. In other words, it is a positive integer that is divisible by numbers other than 1 and itself. The smallest composite number is 4. The first few composite numbers are as follows: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21...... Another way of defining a composite number is "an integer which is exactly divisible by at least one positive integer other than both itself and 1. All numbers are divisible by both 1 and itself. That is, a number which has more than two divisors other than 1 and the number itself is called a composite number.
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Prime Factorizations:
A whole number greater than 1 with precisely two factors, itself and 1, is a prime number. A whole number greater than 1 with above 2 factors is a composite number. The numbers 0 and 1 are neither prime nor composite: 0 has infinite factor, and 1 only has one factor, itself.
The number 2 is the only even prime number. A number that is expressed as a product of factors that are all prime is called the prime factorizations of the number. For example, the prime factorizations of 12 are 2 x 2 x 3.
The prime factorizations of a number is the expression f the number as a product of its prime factors. Begin with the small prime number as a trial divisor, and continue with prime number as trial divisor until the final quotient is 1.
There are two main ways of finding the primes of a number: dividing and splitting.
Properties of Prime Number:
a) 2 is the only even prime number and also lowest even prime number.
b) 3 is the lowest odd prime number.
c) Between 1 and 100 prime numbers are 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. Between numbers 1 to 100 there are 25 prime numbers.
d) In negative numbers there are no prime numbers.
Analyzing the nature of composite numbers:
4 = 2 x 2
6 = 2 x 3
8 = 2 x 4
10 = 2 x 5
12 = 2 x 6, 3 x 4
14 = 2 x 7
15 = 3 x 5
16 = 4 x 4, 2 x 8, 2 x 2 x 2 x 2
18 = 2 x 3 x 3, 2 x 9
Every composite number can be expressed as the product of 2 or more (not necessarily distinct) primes.
The opposite of a composite number is "prime" number. A number is said to be prime if it cannot be divided evenly. Prime numbers are divisible by 1 and itself only. For example, 13 is divisible only by 1 and 13. The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...... The number 1 is neither prime nor composite. The number 2 is the only even prime number. All other prime numbers are odd. All even numbers greater than 3 are composite.
Examples:
Example 1: Check whether the number 3431 is prime or not.
Solution:
Step 1: Find the square root of 3431. The Square root of 3431 is 58.57......
Step 2: If the result of square root is an integer it is automatically composite number.
Step 3: If number ends with 0, 2, 4, 5, 6, 8. Then that is not prime number. 3431 ends in a 1, so go to next step
Step 4: Find the sum of digits of number, if the sum is divisible by 3, given number is composite number -> 3 + 4 + 3 + 1= 11. 11 is not divisible by 3.
step 5: Divide the given number by all the prime numbers less than the square root. Note: you can skip 2, 3, and 5.
Since the square root of 3431 is 58.57... Divide 3431 by primes less than 58
(7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53).
Since 3431 is divisible by 47, it is not prime number and therefore composite number.
Step 6: If a given number is not divisible by any of the prime numbers less than the square root, it is prime number otherwise it is composite number.
Example 2: Check whether the number 2657 is prime or not.
Solution:
Step 1: Find the square root of 2657. The Square root of 2657 is 51.54......
Step 2: If the result of square root is an integer it is automatically composite number.
Step 3: If number ends with 0, 2, 4, 5, 6, 8. Then that is not prime number. 2657 ends in a 7, so go to next step
Step 4: Find the sum of digits of number, if the sum is divisible by 3, given number is composite number -> 2 + 6 + 5 + 7= 20. 20 is not divisible by 3.
Step 5: Divide the given number by all the prime numbers less than the square root. Note: you can skip 2, 3, and 5.
Since the square root of 2657 is 51.54... Divide 2657 by primes less than 51
(7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47).
Since 2657 is not divisible by any of those number, it is prime number and therefore not composite number.
Example 3: Show 1985 and 2878 are prime or composite number.
Solution:
1985 is a composite number since the ending number is with 5. So surely it will be divisible by 5 and hence it is a composite number.
2878 is a composite number since the ending number is with 8. Hence 2878 is an even number and surely it is a composite number.
Example 4: Find the primes of 44 by division.
Solution:
Step 1: To find the primes by division, you must only divide 44 by prime numbers until you can only divide by one.
44 2 = 22, 22 2 = 11, 11 11 = 1, 11 1 = 11 (1 is not prime)
Step 2: All the prime numbers used as divisors make up the prime factorizations of 4.
44 = 2 x 2 x 11
Multiply the prime numbers together, and you should get the original value.
Example 5: Find the prime factorizations of 306.
Solution:
Step 1: To find the primes by division, you must only divide 306 by prime numbers until you can only divide by one.
306 are divisible by 9. 306 = 34 x 9, 9 can be broken down means 3 x 3. So you have 34 x 3 x 3. 34 divisible by 2 that make 17 x 2 x 3 x 3. 17 is a prime number.
Step 2: All the prime numbers used as divisors make up the prime factorizations.
306 = 17 x 2 x 3 x 3
Multiply the prime numbers together, and you should get the original value.
Practice problems:
Practice problem 1:
Find the prime factorizations of 630.
Answer: 2 x 3 x 3 x 5 x 7
Practice problem 2:
Find the prime factorizations of 72.
Answer: 2 x 2 x 2 x 3 x 3
Practice problem 3:
Find the prime factorizations of 42.
Answer: 2 x 3 x 7.
by: Omkar Nayak
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