# The Phi Function

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Introduction

Maths Investigation 1: Phi Function

The Phi Function

The investigation I chose to do is called The Phi Function. Phi Function means “Integers which are less than n and which have no other factor other than 1.” My first task was to work out the value of the Phi Functions:φ(3), φ(8), φ(11) and φ(24). Next I have to make some of my own and find the values of them as well. My second task is to check that if these are correct: φ(7x4) = φ(7) x φ(4) and φ(6x4) = φ(6) x φ(4) then create some of my own and check that if they are equal to each other or not.

## Part 1

(1) φ(3) = 2

The factors of φ(3) are: 1 and 3.

The integers, which are less than 3, are 1 and 2. The table below shows the integers, factors and whether it fits into the expression the number of positive integers less than n which have no factor (other than 1) in common (are co-prime) with n.

Integers | Factors | Does it fit into expression? Yes or No |

1 | 1 | yes |

2 | 1,2 | yes |

Middle

18

1,2,3,6,9,18

No

19

1,19

Yes

20

1,2,4,5,10,20

No

21

1,3,7,21

No

22

1,2,11,22

No

23

1,23

Yes

From looking at the above table I can tell you that the phi function of 8 is 4. This is because there are 4 integers less than 8 whose factors have nothing in common with the factors of 8 except 1.

(5) φ(17) = 16

The factors of 17 are: 1 and 17.

The integers, which are less than 17, are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16. The table below is similar to the one above, but is for the phi function of 17.

Integers | Factors | Does it fit into expression? Yes or No |

1 | 1 | Yes |

2 | 1,2 | Yes |

3 | 1,3 | Yes |

4 | 1,2,4 | Yes |

5 | 1,5 | Yes |

6 | 1,2,3,6 | Yes |

7 | 1,7 | Yes |

8 | 1,2,4,8 | Yes |

9 | 1,3,9 | Yes |

10 | 1,2,5,10 | Yes |

11 | 1,11 | Yes |

12 | 1,2,3,4,6,12 | Yes |

13 | 1,13 | Yes |

14 | 1,2,7,14 | Yes |

15 | 1,3,5,15 | Yes |

16 | 1,2,4,8,16 | Yes |

By looking at the table above, I can tell you that the phi function of 17 is 16 since all integers fit into the expression.

Part 2

(1) φ(7x4) = φ(28) = 12, φ(7) = 6 and φ(4) = 2

The factors of 28 are: 1, 2, 4, 7, 14, and 28. The factors of 7 are: 1 and 7. The factors of 4 are: 1, 2 and 4.

The integers which are below 28 are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 and 27. The integers which are below 7 are: 1, 2, 3, 4, 5 and 6. The integers which are below 4 are: 1, 2 and 3.

Conclusion

1

1

3, 4, 12

2

1,2

3

3

1,3

4

4

1,2,4

None

5

1,5

12

6

1,2,3,6

None

7

1,7

12

8

1,2,4,8

None

9

1,3,9

None

10

1,2,5,10

None

11

1,11

12

By looking at the above table, you can see that the phi function of 3 is 2, the phi function of 4 is 2 and the phi function of 12 is 4. This shows that phi 12 is equal to phi 4 times phi 3. This is because the phi function of 4 which is 2 multiplied by the phi function of 3 which is 2 gives you the phi function of 12 which is 4.

Conclusion

From my investigation of Phi Function, I have done all my investigation in a little space of time, using a brief knowledge of how to work out Phi Function, even though I had no idea at all, what Phi Function was before hand. While I was working out my Phi Function, I had realised a pattern within my coursework. The pattern I discovered was that the Phi Function of a prime number would always be one less than the prime number itself. I also found out a rule for checking out two phi’s to see if they match. The rule is phi(n*m) = phi(n)*phi(m) if n and m are co-prime.

Abdul Thahir Y11

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

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