Lesson 4ª

 

 

 

 

 

   

CALCULATING EVERY PRIME NUMBER THERE IS BETWEEN THE FIRST 209 NATURAL NUMBERS

Eratostenes was a mathematician who was born almost 300 years before Christ. He came up with a simple way of calculating prime numbers. .

ERATOSTENES' SIEVE:
Before we begin, we must understand that sieve is like a tool. It is generally a metal mesh used for sieving or cleaning impurities from wheat and other seeds.
Eratostenes sieve cleans compound numbers leaving prime numbers apart.

Let's do it:

1º we write natural numbers from 1 to 209:

2º We multiply 2 by itself and it gives us 4.

3º We scratch out number 4 (in this example, we have painted its box red).

4º We count two spaces after 4 and we find number 6 and we do the same.

5º we keep counting 2 spaces and painting that box red.
Every number up to now in a red box is NOT a prime number. All of them have 2 as a denominator. This is what we have left:

6º Now, we multiply 3 by 3. we get 9 as a result. We scratch it and we count 3 spots after that (WE ALSO COUNT THOSE WHICH ARE ALREADY MARKED IN RED OR SCRATCHED).

We scratch the ones which aren't scratched. In this case, I have used green. This is what we now have left:

7º Now, we multiply 5 by itself and we get 25.

 We locate number 25, we scratch it, we count 5 spots after that (including every scratched number in red or green) and 5 more. We scratch the numbers in yellow (if they are not already painted in red or green) until we reach the end.

 

8º Now, we multiply 7 by itself. We get 49 so we locate it and we scratch it or paint it. From this number on, we will count seven spots and we will paint them in blue (as before, including the previously painted).

9º We take number 11 and we multiply it by itself. We locate 121 and we start counting 11 spots until we reach the end. We scratch every number we find. We will use a light pink colour this time.

10 º We take number 13 and we multiply it by itself. We get 169. As we have done previously, we count 13 spots from there and we do so until we reach the last number. Remember, we also count those numbers which are already scratched by the prior exercises. We will use a gray colour.

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Number 17 is the number following 13. If we multiply it by itself, we get a number over 209. This means we have finished.

EVERY NUMBER LEFT IN WHITE IS A PRIME NUMBER.

3.17 Try doing a sieve longer than the first 30 prime numbers. If you have a doubt, you can review what you have been explained so far.

MULTIPLES AND DENOMINATORS:
We can say that a number (12) is a multiple of another (4) when after dividing the first number by the second one, the answer is zero (0):

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In this case, 12 is a multiple of 3.

Answer the following questions:

3.17 Is 12 a multiple of 4?

3.18 Is 36 a multiple of 4?

3.19 Is 45 a multiple of 3?

3.20 Is 55 a multiple of 11?

3.21 Is 63 a multiple of 3?

3.22 Is 122 a multiple of 4?

3.23 Is 217 a multiple of 7?

3.24 Is 100 a multiple of 4?

3.25 Is 76 a multiple of 6?

Answers:

3.17 Yes. When we divide them, we get a remainder of zero (0).

3.18 Yes. When we divide them, we get a remainder of zero (0).

3.19 Yes. When we divide them, we get a remainder of zero (0).

3.20 Yes. When we divide them, we get a remainder of zero (0).

3.21 Yes. When we divide them, we get a remainder of zero (0).

3.22 No. When we divide them, we don't get a remainder of zero (0).

3.23 Yes. When we divide them, we get a remainder of zero (0).

3.24 Yes. When we divide them, we get a remainder of zero (0).

3.25 No. When we divide them, we don't we get a remainder of zero (0).

It is said that a number is a denominator of another one when it divides that number in an exact number of times.
Example:             3 divides 9 exactly
                           5 doesn't divide 12 exactly
We can say that 3 is a denominator of 9 and that 5 is not a denominator of 12.

3.26 Is 7 a denominator of 21?
3.27 Is 5 a denominator of 127?
3.28 Is 3 a denominator of 21?
3.29 Is 11 a denominator of 121?
3.30 Is 2 a denominator of 231?
3.31 Is 4 a denominator of 1000?
3.32 Is 3 a denominator of 213?
3.33 Is 6 a denominator of 218?
3.34 Is 7 a denominator of 210?

 

Answers:
3.26     21 is divided an exact number of times by 7. Yes, 7 is a denominator of 21.
3.27    127 is not divided an exact number of times by 5. No, 5 is not a denominator of 127.
3.28    21 is divided an exact number of times by 3. Yes, 3 is a denominator of 21.
3.29     121 is divided an exact number of times by 11. Yes, 11 is a denominator of 121.
3.30     231 is not divided an exact number of times by 2. No, 2 is not a denominator of 231.
3.31     1000 is divided an exact number of times by 4. Yes, 4 is a denominator of 1000.
3.32     213 is divided an exact number of times by 3. Yes, 3 is a denominator of 213.
3.33    218 is not divided an exact number of times by 6. No, 6 is not a denominator of 218.
3.34    210 is divided an exact number of times by 7. Yes, 7 is a denominator of 210.