Number patterns with the Sieve of Eratosthenes

A prime number, you will recall, is divisible only by one and itself.  This is important when it comes to figuring out the lowest common denominator (LCD).


How can we know if a number is prime?  The ancient Greeks had a method to find primes of a given umber, and through the centuies it has become known as the Sieve of Eratosthenes.


This summer fun number exercise about prime numbers is from Mathematics Methods for Elementary and Middle School Teachers, by Hatfield, Edwards, Bitter and Morrow, pub by John Wiley and Sons, New York, (c) 2000

More than 2000 years ago the Greek mathematician Eratosthenes created a process to help sieve (filter out) the composite numbers, leaving only the prime numbers.  There are many interesting patterns to be found by using the Sieve of Eratosthenes.  Teachers can lead students to discover many of the patterns on their own by asking a few leading questions such as the ones found in the problem-solving exploration that follows.

Directions:

1.  You are asked to use six colors on the Sieve of Eratosthenes so that you can see the number patterns more easily.  The colors of yellow, green, pink, light blue, red, and black are suggested, but any eye-catching colors will do.

2.  On the chart in step 4, find the first prime humber, which is 2, and circle it with a red marker.  Now slash (/) all the multiples of 2 using the red marker.  You hve just eliminated all even numbers because they are composites.  Note the color pattern you have just created.

3. Now find the next prime which is 3 and circle it with a green marker.  Sieve (/) all the multiples of 3 using the green marker.  Some numbers now have a red and green slash.  You can easily see the numbers that are multiples of both 2 and 3.  Note the color pattern created by the green slashes.

4. Follow the same procedure using this color code:

Primes            Color
five                 = yellow
seven              = black
eleven             = pink
all others         = light blue


1         2        3        4        5        6        7        8        9       10

11      12     13      14       15     16      17      18      19       20

21      22     23       24      25     26      27      28       29      30

31      32     33       34      35     36      37       38      39       40

41      42     43      44       45     46      47       48      49       50

51      52      53      54      55      56      57       58      59      60

61      62      63      64       65      66     67       68      69      70

71     72       73      74      75       76      77      78       79      80

81     82       83      84       85      86      87      88       89      90

91     92       93      94       95      96      97      98       99    100

Questions based on the Sieve of Eratosthenes

1. How many of the first 100 numbers are prime?

2. Of the first 25 numbers, what percent are multiples of three numbers?  Of the second 25 numbers, what percent are multiples of three numbers?

3. Would you expect this number to increase or decrease as we continue to go through the number system?  Why?

4. The number of primes in the first 50 numbers is what fractional portion of the total primes in the first 100 numbers?  What about the second 50 numbers?

5. Will the relationship found in question 4 hold for the second 100 numbers?  What could be done to find out?

6. When is the first time that a multiple of 7 has not been crossed out by a multiple of a smaller prime?

7. What do you notice about multiples of 11?  Predict at which number the multiple of 11 will be crossed out for the first time with no smaller primes as its multiple.