What is the prime number?
Any Natural number greater than 1, which is divisible by 1 and only by itself is called the prime number.
2, 3, 5, 7, 11 etc.
Arrangement of prime numbers
Prime numbers can be arranged in a line form like shown below in the diagram.
What is a composite number?
Any Natural number, which is greater than 1 and is not a Prime Number is called a composite number.
4, 6, 8, 9, 10 etc.
1 is neither a Prime Number nor Composite Number.
Every number in maths can either be a prime or a composite number except 0 and 1 because 0 and 1 are neither a prime number nor a composite number.
Arrangement of composite numbers
Composite numbers can be arranged in a rectangular or square form like shown as below in the diagram.
How to find prime numbers?
Many mathematicians have worked on primality tests to test whether a number is prime. However, the oldest study of prime numbers was conducted by ancient Greek mathematicians. One of such famous demonstrations was given by Euclid in his book Euclid's Elements, which discusses the infinite many primes.
Fibonacci, who was an Italian mathematician, was the first to describe trial division for primality test. But the Greek mathematician Eratosthenes found a simple method to find prime numbers. This algorithm is known as the Sieve Of Eratosthenes to find prime numbers.
Sieve of eratosthenes
In the sieve of eratosthenes method, each number is tested for its primality, by processing each number one by
one.
Let's understand how this algorithm works by finding the prime numbers between 1 and 100.
Step 1: Cross out 1 as it is neither a prime nor a composite.
Step 2: Encircle 2 and cross out all multiples of 2.
Step 3: Encircle 3 and cross out all multiples of 3, leaving out the one which has been already crossed out.
Step 4: Encircle 5 and cross out all multiples of 5, again do not cross out which have already been crossed.
Step 5: Encircle 7 and cross out all multiples of 7 and do not cross out again, which are already crossed.
So, there are 25 prime numbers between1 and 100.
- A prime number has exactly two divisors.
- 2 is the smallest prime number.
- 2 is also the smallest even prime number.
- All prime numbers except 2 are odd.
- 3 is the smallest odd prime number.
- 4 is the smallest composite prime number.
- 2 and 3 are the only consecutive prime numbers.
- The set of prime numbers is denoted by P.
What are coprimes?
Two numbers are called to be coprime or co-prime if they have only 1 as a common factor.
In other way, we can say if the greatest common divisor of two numbers is 1, then the two numbers are coprimes.
(2,3), (14,15), (8,13) etc.
What is twin prime?
Two numbers are called Twin Prime if they differ by 2 only or we can say that there is only one composite number between them. One prime number can be 2 more or 2 less than another prime number.
(3, 5), (29, 31), (71, 73) etc.
List of twin prime numbers between 1 and 100 are:
- (3, 5)
- (5, 7)
- (11, 13)
- (17, 19)
- (29, 31)
- (41, 43)
- (59, 61)
- (71, 73)
5 is the only prime number that belongs to two pairs (3, 5) and (5, 7).
What are prime triplets?
A set of three consecutive Prime Numbers if the smallest and the largest differ by 6 and must be in the form of (p, p + 2, p + 6) is called a Prime Triplet.
(5,7,11)
Why?
Because, 5, 7 and 11 is in the form of p, p + 2, p + 6
where p = 5
p + 2 = 5 + 2 = 7
and p + 6 = 5 + 6 = 11
-
142
142 is a composite number because it is divisible by 2. -
211
211 is a prime number. -
291
291 is a composite number because it is divisible by 3. -
323
323 is a composite number because it is divisible by 17. -
427
427 is a composite number because it is divisible by 7.
(59, 61) and (71, 73) are the twin primes between 50 and 100.
90, 91, 92, 93, 94, 95, 96
97 is the only prime number between 90 and 100.
2 is the only prime number between 1 and 100.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47 are the prime numbers which are less than 50.
The following pairs are the twin primes between 1 and 100.
(3, 5)
(5, 7)
(11, 13)
(17, 19)
(29, 31)
(41, 43)
(59, 61)
(71, 73)
The total number of prime numbers between 1 and 100 are 25.
- The total number of even prime numbers is ___.
- The smallest prime number is ___.
- The number of primes between 0 and 100 is ___.
- The least odd prime number is ___.
- The pair of (71, 73) is called ___ primes.
- 138 is a ___ number.
- Every prime number except 2 is an ___ number.
- The pair of (14, 15) is called ___.
- Any two ___ numbers are always coprime.
- ___ is neither prime nor composite number.
- Write prime numbers between 90 and 100.
- Write all pairs of twin primes between 1 and 50.
- Write all prime numbers between 10 and 40.
- Find all prime numbers between 70 and 80.
-
Find factors of the following numbers and check whether each number is a prime number.
- 45
- 31
- 99
- 58
- 71
- 91
- 97
- 99
- 93
- 97
- 99
- 93
- 98
- prime
- composite
- twin
- unique
- consecutive
- twin
- unique
- odd
- 2
- 3
- 1
- 4
- 1
- 2
- 5
- 7
- 41 and 43
- 71 and 73
- 53 and 59
- 11 and 13
- 3 and 17
- 17 and 68
- 14 and 16
- 13 and 91
- 8
- 10
- 12
- 6
- 72, 73, 74
- 71, 72, 73
- 71, 74, 79
- 71, 73, 79
