Study Guide: Prime Numbers
What is the Prime Number?
Prime numbers are all integers greater than 1 that have only two factors, the number itself and 1. For example, the only factors of 3 are 1 and 3, so 3 is prime.
The smallest prime number is 2 because it is only divisible by itself and 1. The number 2 also happens to be the only even prime number.
A number that has more than two factors is called a composite number. Such numbers can be made by combining more than two factors. For example, $2 \times 2 \times 3 = 12$, therefore 12 iS a composite number.
Note that prime factors are themselves prime. In the above example, both 2 and 3 are prime numbers.
Finding Prime Numbers Using Factorization
A common way to find prime numbers is by using the Factorization Method.
Step 1: Find all the prime factors of a number.
$$2 \times 2 \times 3 = 12$$
Step 2: Count the prime factors. If the total number of prime factors is greater than two, then the number not a prime number.
$$12 \text{ has } 3 \text{ prime factors, (2, 2, 3) so } 12 \text{ is not prime.}$$
Finding Prime Numbers Using a General Formula
The above method is error prone. We might make a mistake while factoring a large number. Prime numbers can also be found by using a general algebraic formula.
Two consecutive numbers that are both natural numbers and prime numbers are 2 and 3. Apart from 2 and 3, every prime number can be written in the form of $6n + 1$ or $6n - 1$, where $n$ is a natural number.
For example:
\begin{align} 6(1) - 1 &= 5 \\\ 6(1) + 1 &= 7 \\\ 6(2) - 1 &= 11 \\\ 6(2) + 1 &= 13 \\\ 6(3) - 1 &= 17 \\\ 6(3) + 1 & = 19 \end{align}
Finding Larger Prime Numbers Using a General Formula
To find prime numbers greater than 40, use the general formula, $n^2 + n + 41$, where $n$ is a natural number in the set {$0, 1, 2, ….., 39$}
For example:
\begin{align} (0)^2 + 0 + 41 = 41 \\\ (1)^2 + 1 + 41 = 43 \\\ (2)^2 + 2 + 41 = 47 \\\ (3)^2 + 3 + 41 = 53 \\\ (4)^2 + 4 + 41 = 59 \end{align}
Checking for Primes
Warning: The above formulas are great for finding prime numbers, but be careful! Some solutions will not be prime! If not sure, use these checks:
- Other than 2, no even number is prime. Numbers greater than 2 with an even number in the one’s place cannot be prime.
- Multiples of 3 can not be prime. Add all the digits in a number, if the sum is divisible by 3 it is not a prime number.
- Multiples of 5 can not be prime. If there is a 0 or 5 in the one’s place, the number is divisible by 5, and can not be prime.
- Multiples of 7 can not be prime. Double the digit in the one’s place and subtract it from the other digits. If the result is a multiple of 7, the original number is not prime.
Questions
What is the smallest prime number?
What is the largest prime number?
How many prime numbers are there?
What is the only even prime number?
Can you think of a positive integer that is neither prime nor composite?