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Prime and Composite Numbers

A number is said to a prime number if it has only two factors – 1 and itself. A composite number on the other hand, has more than 2 factors.

To check if a number is prime or composite, we use these steps:

  1. Check if the number is divisible by 2. If it is, then it is a composite number. All even numbers are composite numbers, except 2.
  2. If the number is odd, then do the divisibility test for all prime numbers which are less than or equal to the square root of the number. If the number is not a perfect square, we take the closest square number. If it is a perfect square, then the number is composite.
  3. If the number is not divisible by any of these primes, then the number itself is prime

 

Examples of prime and composite numbers:

 

Example 1: Is 25 a prime or a composite number?

25 = 25 x 1 or 5 x 5.

So factors of 25 are {1, 5, 25}

Since 25 has more than 2 factors, it is a composite number

 

Example 2: What about 73?

Now sqrt{73} is close to and less than sqrt{81} = 9.

All primes less than 9 are 2, 3, 5 and 7.

Now, let’s do the divisibility tests for these prime numbers.

  • 73 is not divisible by 2, since it is not even
  • 7 + 3 = 10 is not divisible by 3. So 73 is not divisible by 3
  • 73 does not end with 5, hence it is not divisible by 5
  • (7 x 3) + 3 = 24 is not divisible by 7

Now 73 is less than 81, and it is not divisible by 2, 3, 5 or 7.

Therefore 73 is a prime number

 

Example 3: Now let’s try 91.

Since sqrt{91} is close to and less than sqrt{100} = 10, the primes less than 10 are 2, 3, 5 7, and 9.

Now, let’s do the divisibility tests for these prime numbers.

  • 91 is an odd number, hence not divisible by 2
  • 9 + 1 = 10 is not divisible by 3. So 91 is not divisible by 3
  • 91 does not end with 5 or 0, hence it is not divisible by 5
  • (9 x 3) + 1 = 28 is divisible by 7, so 91 is divisible by 7
  • And finally 91 is not divisible by 9 since 9 + 7 = 16 is not divisible by 9

Since 91 is divisible by 7, its factors are {1, 7, 13, 91}.

And with more than 2 factors, 91 is a composite number.

 

Remember: Prime numbers that differ by 2 are called twin primes. 3 & 5, 17 & 19 are examples of twin primes