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# Prime Numbers

Prime numbers are an important concept in number theory and have fascinated mathematicians for centuries. They are the building blocks of the natural numbers, and every positive integer greater than 1 is either a prime number or can be expressed as the product of prime numbers.

A prime number can be defined as a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. In other words, a prime number cannot be evenly divided by any number other than 1 and itself. This property makes prime numbers unique and important in many areas of mathematics, such as cryptography and number theory.

Prime numbers have many interesting properties and patterns. For example, there are infinitely many prime numbers, and they become less common as numbers get larger. The distribution of prime numbers is also unpredictable and seemingly random, making it a challenging area of study for mathematicians.

## How do you tell a prime number from a non-prime number?

It's easy to tell if a small number, like 12, is not a prime number, because you know that $3×4=12$ . These types of numbers are called composite numbers.

When you have a larger number such as 91 or 97, you have to do a little trial and error by trying to divide in smaller prime numbers in order, starting with $2,3,5,7,11,13$ ... and so on. See the list below.

If any of these prime numbers go into your large number evenly, it's a composite number.

Take the number 91. When you try 2, 3, and 5, they don't go into 91 evenly-they leave a remainder. But when you try $\left(\frac{91}{7}\right)$ , you get 13. Because $7×13=91$ , 91 cannot be a prime number.

What about 97? None of the smaller prime numbers go into 97, so does that mean it's prime? When can we stop? For that, it's helpful to have a list of prime numbers like the one below.

One interesting trick is that if a number has no factors smaller than its own square root, then it must be prime!

If a positive integer n has no factors smaller than its own square root, then any factors it has must be larger than its square root. But if n is not a prime number, then it must have at least one factor that is smaller than its square root. This is because if $n=a×b$ , where a and b are positive integers, then at least one of a or b must be less than or equal to the square root of n.

Therefore, if we test all the integers up to the square root of n and find no factors, then we can conclude that n has no factors other than 1 and itself, and is therefore a prime number.

For example, to test if 17 is a prime number, we can check all the integers up to the square root of 17, which is approximately 4.12. We can see that 17 is not divisible by any of the integers 2, 3, or 4, so we can conclude that 17 is a prime number.

## Properties of prime numbers

Some of the properties of prime numbers include:

• Every number greater than 1 can be divided by at least one prime number.
• Every even, positive integer greater than 2 can be expressed as the sum of two primes (this statement is so far unproven).
• Other than 2, all prime numbers are odd. 2 is the only even prime number that exists.
• Every composite number can be factored into prime factors. Each of these prime factorizations are unique.

## List of prime numbers from 1 – 100

Prime Numbers Table

Range Prime Numbers
1-10 2, 3, 5, 7
11-20 11, 13, 17, 19
21-30 23, 29
31-40 31, 37
41-50 41, 43, 47
51-60 53, 59
61-70 61, 67
71-80 71, 73, 79
81-90 83, 89
91-100 97

## Flashcards covering the Prime Numbers

Pre-Algebra Flashcards