**What is a prime number?**

In mathematics, we say a natural number $p$ is a prime number if p has exactly 2 factors. For example, 11 is a prime number, because the only factors are 1 and 11. It is easy to see that the only prime numbers less than 10 are 2, 3, 5 and 7. The smallest prime number is 2. The only even prime number is also 2. Another equivalent definition of prime number is a natural number greater than 1 that is not a product of two natural numbers greater than 1. For example, 10 is not a prime number, because $ 10=2\times 5$.

**What is a composite number?**

If a number greater than 1 is not prime, we call it a composite number. The smallest composite number is 4.

**Is 1 a prime number?**

Note than 1 is not a prime number and it is not a composite number.

**List of prime numbers 1 to 100:**

In total we have 25 prime numbers less than 100. Here is the list:

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.

There are a total of 168 prime numbers less than 1000.

**How many prime numbers are there?**

There are infinitely many prime numbers. There are many proofs for this statement. Let’s prove it here with the most famous method:

On the contrary, let’s assume we have finitely many prime numbers, let’s call them $p_1,…,p_k$. Let $n=p_1\times p_2\times … p_k +1$, it is not difficult to see than $n$ is greater than all $p_{1},…,p_{k}$, so $n$ is not one of them, and then, $n$ is a composite number. So $n$ has a prime factor, let’s say $p_i$ is a factor of $n$. It means than $\dfrac{n}{p_i}$ is an integer, but $\dfrac{n}{p_i}=\dfrac{ p_1\times p_2\times … p_k }{p_i} +\dfrac{1}{p_i}=p_1\times p_2\times p_{i-1}\times p_{i+1}\times … p_k +\dfrac{1}{p_i}$, which is not an integers, that contradicts our assumption that we have finitely many prime numbers. So, it proves than There are infinitely many prime numbers.

**How to check if a number is prime?**

To see if a number $n>2$ is a prime number first we check if $n$ is divisible by 2, if it is divisible by 2, then is it not a prime number, then we check divisibility by 3, and then by 5,… . We don’t need to continue to $n$. It is enough to check until $\sqrt{n}$.

For example, to check the number 53, we check this test until $\sqrt{59}$. So we only need to check this test for number, 2, 3, 5 and 7.

**What is prime factorization?**

“*Prime Factorization*” is factoring our number $n$ into prime factors (prime numbers multiply together to make $n$. For example, the prime factorization of 100 is $100=2^{2}\times 5^{2}$. Prime factors of a number $n$ are prime numbers that are multiplied together to get $n$. As an example, 2 and 5 are prime factors are 10. Note that *Prime Factorization is unique.*

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