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Learning Math Home
Number and Operation Session 6: Number Theory
 
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Session 6, Part B:
Looking for Prime Numbers (45 minutes)

In This Part: Locating Prime Numbers | Necessary and Sufficient Conditions
Is This Number Prime?

In this part, we'll continue a mathematics tradition begun by Eratosthenes of Cyrene (276-194 B.C.E.) -- the same person who's known for accurately estimating the diameter of the Earth based on the shadow cast by the Sun's light.

Eratosthenes worked out a method, now called the "Sieve of Eratosthenes," to collect all the prime numbers and allow all composites (multiples of prime numbers) to "drain through." He used a grid that looked like what we now call the 100 board -- the first row is 1-10, the second row 11-20, etc. This grid does locate the prime numbers, but it does not help us understand where to look for them. If you try looking for prime numbers in this grid, you will discover that it's not so easy to locate them in a systematic way:

In the following activity, we will use a different grid to locate the prime numbers. This grid has only six columns, starting with the numbers 2 through 7. As you will see, such positioning of numbers will make the patterns more noticeable and consequently will be more helpful in answering the question of where the prime numbers are located.

Copy the grid above or print a PDF version.

Problem B1

Solution  

Circle the 2, which is a prime number. Next, cross out all the multiples of 2, as they are not prime numbers.

a. 

Imagine that the grid goes on forever. Present a convincing argument for the fact that all the numbers in the first, third, and fifth columns are multiples of 2 and would thus be crossed out.

b. 

Could multiples of 2 be located in any other column?


 

Problem B2

Solution  

Next, circle the smallest remaining prime number (i.e., 3). Cross out all the multiples of 3, as they are not prime.

a. 

Again imagine that the grid goes on forever. Present a convincing argument for the fact that all the numbers in the second and fifth columns are multiples of 3 and would thus be crossed out. (Of course, the fifth column is already gone because it contains multiples of 2!)

b. 

Could multiples of 3 be located in any other column?


 

Problem B3

Solution  

Again, circle the smallest remaining prime number (i.e., 5) and cross out all the multiples of 5, as they are not prime. Notice that these multiples are not all located in particular columns, so crossing them out is not as easy as before. Continue in a similar manner until all the numbers on the grid are either circled or crossed out.

Examine the grid and imagine that it extends to infinity. Where on the extended grid should you look to find prime numbers greater than 3?



video thumbnail
 

Video Segment
Watch how Professor Findell and the participants worked on Problems B1-B3 in this video segment.

Are you convinced that the conjecture about locating prime numbers is true? Try extending the table by a few more rows to see if the rule still applies.

If you are using a VCR, you can find this segment on the session video approximately 10 minutes and 56 seconds after the Annenberg Media logo.

 

Next > Part B (Continued): Necessary and Sufficient Conditions

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