# Prime Numbers

### What is a Prime Number?

A prime number is a positive integer having exactly two factors. If p is a prime, then it’s only factors are necessarily 1 and p itself.

The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

It should be noted that 1 is a non-prime number.

### List of prime numbers from 1 to 100

The table below shows the list of prime numbers from 1 to 100.

### Why 1 is not a prime number?

Conferring to the definition of prime number, which states that a number should have exactly two factors, but number 1 has one and only one factor. Thus 1 is not considered as a Prime number.

### Smallest prime number

The smallest prime number as defined by modern mathematicians is 2. To be prime, a number must be divisible only by 1 and the number itself.

Largest prime number

As of January 2016, the largest known prime number is $2^{77,232,917} – 1$, a number with 23,249,425 digits. It was found by the Great Internet Mersenne Prime Search (GIMPS).

How to find prime numbers:

Let us choose any random number “A.” We can find whether this number is a prime number or not using the following method-

1. Find the number k, where $k^{2}>A$
2. If A is divisible by any prime number less than k, then A is a non-prime number,otherwise A is a Prime number.
 Question – Is 10 a Prime Number? Answer – No, because it can be divided evenly by 2 or 5 , 2×5=10, as well as by 1 and 10. Question – Is 19 a Prime Number? Answer – First we need to find the number k, such that $k^{2}>19$. Thus the value of k comes out to be 5, as $5^{2}=25$.. The prime numbers less than 5 are 2 and 3. Clearly, we can see that 19 is not divisible by both 2 and 3. Therefore, 19 is a prime number.

Note- An Integer P > 1 is termed as a prime number when its only divisors are 1 and P.

Any numeric m > 1 which is not a prime is termed as a composite.

Each composite number can be factored into prime factors, and individually all of these are unique in nature.

Some of the properties of primes are:

• Every n > 1 is can be divided by some prime.
• The product is divisible by a prime ‘p’ only when ‘p’ divides one of the factors.
• A prime ‘p’ is one or the other relatively prime to a number ‘n’ or divide it.
 Example: Find if 53 is a prime number or not ? Solution: The only factors of 53 are 1 and 53. So, 53 is a prime number. Example 2: Check if 64 is a prime number or not? Solution. The factors of 64 are 1, 2, 4, 8, 16, 32, 64. Hence it is a composite number and not a prime number.

#### Practise This Question

Find the value of sinn1890+cosecn1890. Where n ϵ N