Prime numbers
Introduction
A prime number is defined as any natural number with only two factors that are $$1$$ and itself. The number $$2$$ is an example of a prime number. This is because the only way of denoting $$2$$ as a product is $$2 \times 1$$ or $$1 \times 2$$. Also, $$2$$ is the only even prime number.
The first $$10$$ prime numbers are $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, $$17$$, $$19$$, $$23$$, $$29$$.
Some of the important properties related to prime numbers have been stated below.
Every number that is greater than $$1$$ will be divisible by at least one prime number.
Only common factor of any two prime numbers is $$1$$, therefore prime numbers are always co-prime.
A positive integer that is greater than $$2$$ can be expressed in the form of a sum of two primes.
E1.1: Identify and use prime numbers
Composite numbers:
The numbers that can be obtained by taking the product of two smaller positive integers are known as composite numbers. In other words, a number which has three or more factors is a composite number. For example, $$6$$ is a composite number. We can obtain $$6$$ by multiplying $$6$$ by $$1$$ and also by multiplying $$2$$ by $$3$$.
Therefore, $$6$$ has more than two factors, $$1$$, $$2$$, $$3$$ and $$6$$, so it is not a prime number. Note that the numbers which are not prime, except $$1$$, are composite numbers.
Identifying Prime Numbers:
There are two methods to find out if a given number is prime.
Except $$2$$ and $$3$$, every prime number can be denoted in the form of $$6n+1$$ or $$6n-1$$ (not including the multiples of prime numbers, that is, $$2$$, $$3$$, $$5$$, $$7$$, $$11$$) where $$n$$ is any natural number. For example,
$$6(1) – 1 = 5$$
$$6(1) + 1 = 7$$
$$6(2) – 1 = 11$$
$$6(2) + 1 = 13$$
$$6(3) – 1 = 17$$
$$6(3) + 1 = 19$$
$$6(4) – 1 = 23$$
$$6(4) + 1 = 25$$ (Multiple of $$5$$)
We can use the formula, $$n^2+n+41$$ to obtain the prime numbers that are greater than $$40$$. Here, $$n$$ can be a number between $$0$$, $$1$$, $$2$$, $$3$$, …, $$39$$. For example,
$$0^2+0+41 = 41$$
$$1^2+1+41 = 43$$
Mathematical description of prime numbers:
Each prime number is a positive number which cannot be expressed as a product of two even integers. There are a number of primes in the number system. Following is the table of all-natural numbers from $$1$$ to $$100$$. The dark boxes represent the prime numbers.
Worked examples
Example 1: Is $$10$$ a prime number?
Step 1: Find the factors of $$10$$.
The factors of $$10$$ are $$1$$, $$2$$, $$5$$ and $$10$$.
Step 2: write answer according to the definition of prime numbers
Since, $$10$$ has more than two factors. Therefore, the number $$10$$ is not a prime number.
Example 2: Is $$17$$ a prime number?
Step 1: Find the factors of $$17$$.
The factors of $$17$$ are $$1$$ and $$17$$.
Step 2: write answer according to the definition of prime numbers
Since, the number $$17$$ has only two factors, $$1$$ and $$17$$. Therefore, it is a prime number.
All prime numbers except 2 can be written as a sum of two primes. The only even prime number is 2. Natural number 1 is neither prime nor composite. On dividing a prime number with any other number besides 1 and the number itself the remainder is always not equal to zero. Any two prime numbers are always co prime to each other.