# The Phi Function Investigation

Extracts from this document...

Introduction

The Phi Function For any positive integer n, the Phi Function ?(n) is defined as the number of positive integers less than n which have no factor (other than 1) in common (are co-prime) with n. Part 1 (a) Find the value of: (I) ?(3) (ii) ?(8) (iii) ?(11) (iv) ?(24) (b) Obtain the Phi-Function for at least 5 positive integers of your own choice. (a) (I) ?(3): 1 1 2 1,2 3 1,3 3 = 1,2 The number 3 only has 2 positive integers they are the numbers 1 and 2. (ii) ?(8): 1 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 8 = 1,3,5,7 There are 4 positive integers for the number 8 (iii) ?(11): 1 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 9 1,3,9 10 1,2,5,10 11 1,11 11 = 1,2,3,4,5,6,7,8,9,10 The number 11 has 10 positive integers, they are shown above. (iv) ?(24): 1 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 9 1,3,9 10 1,2,5,10 11 1,11 12 1,2,4,6,12 13 1,13 14 1,2,7,14 15 1,3,5,15 16 1,2,4,8,16 17 1,17 18 1,2,6,9,18 19 1,19 20 1,2,4,5,10,20 21 1,3,21 22 1,2,11,22 23 1,23 24 1,2,3,4,6,8,12,24 24 = 1,5,7,11,13,17,19,23 The number 24 has 8 positive integers, they are shown above. ...read more.

Middle

?(24): 1 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 9 1,3,9 10 1,2,5,10 11 1,11 12 1,2,6,12 13 1,13 14 1,2,7,14 15 1,3,5,15 16 1,2,4,8,16 17 1,17 18 1,2,6,9,18 19 1,19 20 1,2,4,5,10,20 21 1,3,21 22 1,2,11,22 23 1,23 24 1,2,3,4,6,8,12,24 24 = 1,5,7,11,13,17,19,23 The number 24 has 8 positive integers, they are shown above. ?(6): 1 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 6 = 1 and 5 The number 6 has 2 positive integers, they are shown above. ?(4): 1 1 2 1,2 3 1,3 4 1,2,4 4 = 1 and 3 The number 4 has 2 positive integers, they are shown above. The number 24 has 8 positive integers and the numbers 6 and 4 each have 2 positive integers each. When they are multiplied together they equal 4. This does not equal 8 so the equation is correct as shown in the box below. (b) Check whether or not ?(n � m) = ?(n) � ?(m) for at least two separate choices of n and m. The first two choices I will use will consist of the number 3 for the letter n and the number 9 for the letter m. ...read more.

Conclusion

I have chose these two numbers as they do not have common factors. ?(4): 1 1 2 1,2 3 1,3 4 1,2,4 4 = 1 and 3 The number 4 has 2 positive integers, they are shown above. ?(4): 1 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 9 1,3,9 9 = 1,2,4,5,6,7,8 The number 9 has 7 positive integers, they are shown above. ?(36): 1 1 2 1,2 3 1,3 4 1,2,4 5 1,5 6 1,2,3,6 7 1,7 8 1,2,4,8 9 1,3,9 10 1,2,5,10 11 1,11 12 1,2,3,4,6,12 13 1,13 14 1,2,7,14,28 15 1,3,5,15 16 1,2,8,16 17 1,17 18 1,2,6,9,18 19 1,19 20 1,2,4,5,10,20 21 1,3,7,21 22 1,2,11,22 23 1,23 24 1,2,3,6,8,12,24 25 1,5,25 26 1,2,13,26 27 1,3,9,27 28 1,2,4,14,28 29 1,29 30 1,2,3,5,10,15,30 31 1,31 32 1,2,4,8,16,32 33 1,3,11,33 34 1,2,17,34 35 1,5,7,35 36 1,2,3,9,12,18,36 36 = 1,5,7,11,13,17,19,23,25,29,31,32,35,36 The number 36 has 14 positive integers, they are shown above. The number 4 produced 2 positive integers and the number 9 produced 7 positive integers. When multiplied together this equalled the amount of positive integers the number 36 produced. (Which was coincidentally 14) This is another example of the formula that states that: "Numbers that do not have common factors work in the equation ?(n � m) = ?(n) � ?(m)" ...read more.

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