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• Level: GCSE
• Subject: Maths
• Word count: 1955

# The Phi Function

Extracts from this document...

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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# Related GCSE Phi Function essays

1. ## Identify and explain the rules and equations associated with the Phi function.

For example ?32 (also wrote as 25) =24 Phi values of 3 This has led me onto my next number which is 3. ?3=2 ?21=12 ?6=2 ?24=8 ?9=6 ?27=18 ?12=4 ?30=8 ?15=8 ?33=20 ?18=6 ?36=12 All these numbers are divisible by 2 I again did not see any clear pattern emerging.

2. ## Investigating the Phi function

= 2 (8) = 4 (16) = 8 (32) = 16 (and the trend carries on) This is the same for when you start the very beginning of the trend with the phi numbers (6) and(10). Their answers are proportional: if you double the phi number the answer is doubled.

1. ## The totient function.

Eg: ?7= 6 This means that we can say: ?Prime (n) = n - 1 This formula is true for prime numbers because they have only 1 as a common factor, therefore all the numbers less that (n) will be co-prime to it.

2. ## The phi function.

because its cubed Solution 1- ? (5) = 4, 3, 2, 1. ? (5) = 4 2- ? (10) =9, 7, 3, 1. ? (10) = 4 3- ? (15) = 14, 13, 12, 11, 8, 7, 4, 2, 1.

1. ## The Phi function.

So we can see that the difference has to multiplied by 2. This means that the value of the latter odd number that is not a prime number will be more than that of the preceding odd number that is not prime.

2. ## In this coursework I was asked to investigate the Phi Function (f) of a ...

(22); 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21; =10 ? (23); 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22; =22 ?

1. ## Mathematics GCSE Coursework - The Phi Function.

x ?(3)=6 x 2=12; So ?(7) x ?(3)=6 x 2=12 is true. ?(4 x 5)=?(4) x ?(5); ?(4)=2; I have found ?(4) already. ?(5)=4; I have found ?(5) already. ?(4 x 5)= ?(20)=8; 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ?(4)

2. ## Maths Primes and Multiples Investigation

?(3x4) = ?(3) x ?(4) ?(3)=2 ?(4)=2 2x2=4 ?(12)=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11=4 4=4, therefore a prime and a non-prime work sometimes. To finish my non-prime column I will try two non-primes now: ?(4x6) = ?(4) x ?(6) ?(6)=2 ?(4)=2 2x2=4 ?(24)=8 4=8, therefore two non-primes don't work.

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