Whole numbers are comprised of two types, primes and composites.  Primes are any numbers that are only divisible by the number one and themselves.  Composite numbers have three or more factors: one, themselves, and other factor(s).

The following numbers are primes:  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, etc.…
The following numbers are composite, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, etc.…

The number 1 is a special case.  It qualifies as neither a prime nor a composite. Find out what the number 1 is called.

How do we test if a relatively large number is prime?
Suppose n is a whole number, and we want to test it to see if it is prime.   First, we take the square root (or the 1/2 power) of n; then we round this number up to the next highest whole number.  Call the result m.  Next we make a list of all the prime, whole numbers that are less than m. We must divide n by every prime, whole number in our list. If one of the prime, whole numbers in our list evenly divides into n, then n is composite. If none of the prime, whole numbers in our list divides into n, then n is prime. This is known as the prime number test.

A computer chart to find prime numbers less than 200:
Sieve of Eratosthenes
Go to the web site above and find the primes less than 200.

How do we test if a very large number is prime?
Well, we can have the computer do that for us: CLICK HERE.

A factor is simply a number that is multiplied to get a product. Factoring a number means taking the number apart to find its factors or divisors. Here are lists of all the factors of 16, 20, and 45.

          16 --> 1, 2, 4, 8, 16
          20 --> 1, 2, 4, 5, 10, 20
          45 --> 1, 3, 5, 9, 15, 45

Practice finding factors on the following site.

Practice finding factors by making rectangular arrays.

A multiple of a whole number is the product of the whole number and any natural/counting number. For example, to find the multiples of 4, multiply 4 by 1, 4 by 2, 4 by 3, and so on.

Practice finding multiples on the following site: Pascal's Triangle.

When a composite number is written as a product of all of its prime factors, we have the prime factorization of the number. For example, the number 72 can be written as a product of primes as: 72 =2³ x 3². The expression "2³ x 3²" is said to be the prime factorization of 72. The Fundamental Theorem of Arithmetic states that every composite number can be factored uniquely (except for the order of the factors) into a product of prime factors.
Practice Factor Trees: CLICK HERE.

Imagine you are at a school that still has student lockers. There are 1000 lockers, all shut and unlocked, and 1000 students.
Here's the problem:
1.Suppose the first student goes along the row and opens every locker.
2.The second student then goes along and shuts every other locker beginning with locker number 2.
3.The third student changes the state of every third locker beginning with locker number 3. (If the locker is open the student shuts it, and if the locker is closed the student opens it.)
4.The fourth student changes the state of every fourth locker beginning with locker number 4. Imagine that this continues until the thousand students have followed the pattern with the thousand lockers. At the end, which lockers will be open and which will be closed? Why?

Solve this problem and turn in a solution the next time we meet in class. A hint for solving: a spreadsheet might come in real handy!

How can you tell a number is divisible by 2? For example: is 235 divisible by 2?
Write a general rule how you can tell if a number is divisible by 2.
Do the same for 5, 10, then 3 and 9, then 4 and 8.
Bring your rules to class next time!

Updated: 08/21/2003
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