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
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What is the smallest prime number?
The smallest prime number is $2$. -
What is the largest prime number?
No one knows. -
How many prime numbers are there?
No one knows. It is possible that at some point all very larger numbers become composite, or not. -
What is the only even prime number?
The only even prime number is $2$, because all other even numbers have $2$ as a factor. -
Which two prime numbers are consecutive?
The only consecutive prime numbers are $2$ and $3$, because all other prime numbers are odd numbers. -
Is there a positive integer that is neither prime nor composite?
Yes, the number $1$.