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Mathematics GCSE Coursework - The Phi Function.

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Introduction

Mathematics GCSE Coursework

The Phi Function

In this coursework I will be investigating the Phi function.  And I am making clear that crossed numbers like 1 are co-prime, but numbers in circle like 1 are not.

Part 1

a)

Ф(3)=2;

        1 2 3

Ф(8)=4;

        1 2 3 4

Ф(11)=10;

        1 2 3 4 5 6 7 8 9 10 11

Ф(24)=8;

        1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

And I will do five more examples to show my working.

b)

Ф(19)=18;

        1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Ф(9)=6;

        1 2 3 4 5 6 7 8 9

Ф(13)=12;

        1 2 3 4 5 6 7 8 9 10 11 12 13

Ф(5)=4;

        1 2 3 4 5

Ф(15)=8;

        1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Part 2

a)

1)

Ф(7 x 4) = Ф(7) x (4);

        Ф(7)=6;

                1 2 3 4 5 6 7

        Ф(4)=2;

                1 2 3 4

        Ф(7 x 4)= Ф(28)=12;

                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

        Ф(7) x Ф(4)

...read more.

Middle

        Ф(6) x Ф(4)=2 x 2=4;

So Ф(6 x 4)=Ф(6) x Ф(4) is true.

b)

Ф(5 x 2)=Ф(5) x Ф(2);

        Ф(5)=4;

        I have found Ф(5) already.

        Ф(2)=1;

                1 2

        Ф(5 x 2)= Ф(10)=4;

                1 2 3 4 5 6 7 8 9 10

        Ф(5) x Ф(2)=4 x 1=4;

So Ф(5 x 2)=Ф(5) x Ф(2) is true.

Ф(7 x 3)=Ф(7) x Ф(3);

        Ф(7)=6;

        I have found Ф(7) already.

        Ф(3)=2;

        I have found Ф(3) already.

        Ф(7 x 3)= Ф(21)=12;

                1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

        Ф(7) x Ф(3)=6 x 2=12;

So Ф(7) x Ф(3)=6 x 2=12 is true.

Ф(4 x 5)=Ф(4) x Ф(5);

        Ф(4)=2;

        I have found Ф(4) already.

        Ф(5)=4;

        I have found Ф(5) already.

        Ф(4 x 5)= Ф(20)=8;

                1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

        Ф(4) x Ф(5)=2 x 4=8;

So Ф(4 x 5)=Ф(4) x Ф(5) is true.

As you can see I have checked three separate choices of n and m for

Ф(n x m)=Ф(n) x Ф(m), and all of them were true.

...read more.

Conclusion

The prime numbers for 12 are 2 and 3.

And it explains my method of finding a ф of a number. For example I know that prime numbers of 12 are 2 and 3, so I write down 12 numbers, then I cross every second and third number, and the number of not crossed numbers is an answer.

So, for a number, which consists of 2 only I would cross each second numbers which is a half of all numbers, so the other half is the answer.

Part 3

In this part I am going to investigate a rule, which is Ф(n x m)= Ф(n) x Ф(m).

Fist of all I will try to investigate it with odd and even numbers.

Even-Even.

...read more.

This student written piece of work is one of many that can be found in our GCSE Phi Function section.

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