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Prime Numbers- Counting numbers other than one that are divisible only by 1 and themselves.
Composite Numbers- Counting numbers, other than 1, that are not prime.
“Eratosthenes Biography - Math Pioneers Series.” Mr.Nussbaum Learning and Fun, mrnussbaum.com/eratosthenes-biography-math-pioneers-series. Accessed 9 May 2025.
Beckmann, S. (2022). Mathematics for elementary and middle school teachers: With activities (6th ed.). Pearson Education, Inc.
Sieve of Eratosthenes
Sieve of Eratosthenes
Eratosthenes was a mathmititian and astronomer in ancient Greece (275–195 b.c.), who discovered a way to count prime numbers. To use the method, list all the whole numbers from 2 up to wherever you want to stop looking for prime numbers. Then follow the following steps:
1. On the list, circle the number 2, and then cross out every other number after 2. (So cross out 4, 6, 8, etc., the multiples of 2.)
2. Then circle the next number that has not been crossed out, 3, and cross out every third number after 3—even if it has already been crossed out. (So cross out 6, 9, 12, etc., the multiples of 3. Notice that you can do this “mechanically,” by repeatedly counting, “1, 2, 3,” and crossing out on “3.”)
3. Circle the next number that has not been crossed out, 5, and cross out every fifth number after 5—even if it has already been crossed out. (So cross out 10, 15, 20, etc., the multiples of 5. Again, notice that you can do this “mechanically,” by repeatedly count-ing, “1, 2, 3, 4, 5,” and crossing out on “5.”)
4. Continue in this way, going back to the beginning of the list, circling the next number N that has not been crossed out and then crossing out every Nth number after it until every number in the list has either been circled or crossed out.
The circled numbers in the list are the prime numbers.
Trial Division
How can you find out if a number is prime if it's a number such as 149? Using the Sieve of Eratosthenes may take a while. Another method is called trial division. You take your counting number, in this case 149 and begin dividing it by prime numbers... 2, 3, 5, 7, and so on. Your number is prime if at any time the number is divided equally. You know when to stop when your quotient is smaller then the number you are dividing by. In this case 149/13=11.462. 11.462 is smaller than 13 so I know that I can stop dividing and that 149 is not prime.
Factor Tree
A factor tree is another way to factor a counting number into a product of prime numbers. For example beginning with the number 36 you can factor it as such 36 = 2 x 18. Continue on and record the factorizations in order as show below the image to the right. If two people made their factor trees different, they would still end up with the same list of prime numbers!
Image: “What Is a Factor Tree? Definition, Steps, Examples, FAQ, Facts.” SplashLearn, 30 May 2024, www.splashlearn.com/math-vocabulary/factor-tree.
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