*lmost *all of the “counting numbers” (1, 2, 3, 4, 5, etc.) can be placed into one of two categories: prime or composite. **Prime** numbers have __exactly__ two positive factors – one and themselves. **Composite** numbers have more than two positive factors.

## The Number 1:

I say* almost *because 1 fits in neither of these two categories! By definition, 1 is not prime nor is it composite.

A prime number must have __exactly__ two positive factors. 1 has only one factor (which is 1) so 1 is not a prime number. A composite number has to have more than two positive factors. Again, 1 does not meet that requirement with its singular factor so 1 cannot be a composite number. 1 therefore, is a special case. It is the only “counting number” that is neither prime nor composite.

## Even and Odd Primes:

*Most* prime numbers are odd: 3, 5, 7, 23, etc. These are called **odd primes**.

2 however, an exception to the rule. The number 2 is the only even prime number. 2 has exactly two positive factors (2 and 1), therefore it is a prime number. All other even numbers, *also *have 2 as a factor which means they have at least three positive factors and are therefore not prime.

Take 6 for example. The positive factors of 6 are 1, 6, 2 and 3. Having four positive factors makes 6 a composite number.

How about 10? The positive factors of 10 are 1, 10, 2 and 5. Again, having four positive factors makes 10 a composite number.

Every single even number larger than 2 has at least four positive factors. These factors are always 1, the number itself, 2 and its partner. Many even numbers have even more! Therefore, you can know that *all *even numbers larger than 2 are composite numbers.

## The Sieve of Eratosthenes:

By applying the divisibility rules to a hundred chart we can determine which of the numbers between 1 and 100 are prime. This method was developed many years ago by a mathematician and still today bears his name.

- Circle 2, because it IS a prime number.
- Cross out all even numbers
*except*for two. If the number is even (and the number is not 2), the number must be composite so it is NOT prime. - Circle 3. It IS prime.
- Cross out all other numbers divisible by three. Not sure? Add the digits together – if the sum is divisible by three then so is the number. And if the number is divisible by three it is composite and therefore NOT prime.
- Cross out all numbers divisible by four. You can find out if a number is divisible by four by using the divisibility rule of four. If the last two digits are divisible by four than the number is. And if the number is divisible by four it’s NOT prime.
- Circle 5. 5 IS a prime number.
- Cross out all numbers divisible by five. Is the last digit a zero or a five? If so, the number is divisible by five and is NOT prime.
- Cross out all numbers divisible by six. (
*Hint, these should all be crossed out already.)*If the number is even*and*divisible by three it is divisible by six and therefore NOT prime. - Circle 7. 7 IS prime.
- Cross out all numbers divisible by seven. No good trick here, but go through the last remaining numbers to check. Remember, if the number is divisible by seven it is NOT prime.
- Now, circle all of the remaining numbers.

On a fresh chart, color all of the crossed out numbers one color and all of the circled numbers another. Give 1 a color of its own. On our chart, we colored the prime numbers yellow and left the composite numbers white. We made 1 blue.

You can see that there are 25 prime numbers between 1 and 100. There are 74 composite numbers and then of course, there’s 1 which is neither prime nor composite. Did you chart match ours? Nicely done! This method of finding prime numbers is known as the Sieve of Eratosthenes and is named after a famous mathematician.

## How Many Prime Numbers Are There?

As we found above, there are 25 prime numbers between 1 and 100, but that’s just the beginning. Mathematicians tell us that there are actually an **infinite **number of prime numbers in the universe!

## Large Prime Numbers

While many prime numbers are small (2, 3, 5, etc.) some prime numbers are very large! Although calculations tell us that prime numbers are less and less likely to be prime the larger the numbers get some very large prime numbers have been discovered. In fact, the largest prime number ever discovered contains more than 20,000 digits.

## Prime Numbers in the Real World

Prime numbers are far more than just a cool math thing to know. Prime numbers are used daily by encryption specialists to secure some of the most valuable assets in the world. By multiplying two large prime numbers together, we are able to achieve an extremely large and *nearly* un-factorable number to be used in encryption.

Right now, prime numbers are also being used to establish the color intensity of pixels on your screen.

Prime numbers are even found in nature. Cicadas, for example, time their life cycles based on prime numbers.

The bottom line is, that whether or not you know it, you depend on prime numbers each and every day.

## Article comments

I love it when real life examples of how math we learn is used every day! That is so important because it brings textbook to life . Thank you!