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Maths Driving test data: analysis and manipulation.
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Introduction The objective of this project is to successfully manipulate a large set of data in order to prove/ disprove a set of hypotheses. The data consists of results and statistics from a set of driving instruction and tests. The dependant variable in the data is the number of mistakes made during the test. The hypotheses will have been synthesized by myself and in response to influenced by the data that I have been given. The data will have to be subjected to sampling to reduce the vast amount of information. The data will then be processed so...

M 35 5 C Wed 14

M 40 4 C Thur 13

F 10 37 D Thur 10

F 17 31 D Wed 13

F 24 28 D Mon 10

F 31 24 D Mon 17

F 32 17 D Fri 14

F 40 19 D Fri 17

M 15 35 D Wed 14

M 25 28 D Thur 9

M 28 26 D Tue 12

M 29 27 D Thur 12

M 40 4 D Fri 14

M 40 20 D Wed 16

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Introduction In this investigation I...Introduction In this investigation I have been given the task of discovering a formula to tell us the value of n when 'n' is any given number. Also, in the second part of my investigation, look into whether there is a link between the phi of a number, and the product of its factors' phis. Such as: Is it true that 6 x 4 = 6 x 4? How to Find the Phi of a Number For any positive integer n, the phi function n is defined as the number of positive integers les than n which have no factor other than 1 in common are co-prime with n. So 10 = 4 because the positive integers less than 10 which have no factors other than 1, in common with 10 are 1, 3, 7 and 9 i.e. 4 of them. Also 16 = 8 because the integers less than 16 which have no factors other than 1, in common with 16 are 1, 3, 5, 7, 9, 11, 13 and 15 i.e. 8 of them. Part 1 i 3 = 1, 2 = 2 ii 8 = 1, 3, 5, 7 = 4 iii 24 = 1, 5, 7, 9, 11, 13, 17, 19, 24 = 9 iv 11 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 = 10 To investigate this idea further, I will now find the phi functions for the numbers up to, and including 20, to see if I can find any patterns which will indicate to me a formula for n. ** In this investigation if the phi of any number has already been studied, only the phi will be given, without any working. ** ** Also only the numbers which are not factors of the phi in question will be stated. ** 1 = 0 2 = 1 = 1 3 = 1, 2 = 2 4 = 1, 3 = 2 5 = 1, 2, 3, 4 = 4 6 = 1, 5 = 2 7 = 1, 2, 3, 4, 5, 6 = 6 8 = 1, 3, 5, 7 = 4 9 = 1, 2, 4, 5, 7, 8 = 6 10 = 1, 3, 7, 9 = 4 11 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 = 10 12 = 1, 5, 7, 11 = 4 13 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 = 12 14 = 1, 3, 5, 9, 11, 13 = 6 15 = 1, 2, 4, 7, 8, 11, 13, 14 = 8 16 = 1, 3, 5, 7, 9, 11, 13, 15 = 8 17 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 = 16 18 = 1, 5, 7, 11, 13, 17 = 6 19 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 = 18 20 = 1, 3, 7, 9, 11, 13, 17, 19 = 8 Sequence: 0, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8. Summary From studying the phi of numbers 1-20 there is clearly no outstanding link between the numbers, and their phi's. Although I can draw several points from this: Prime Numbers The prime numbers from the previous sequence were: 2, 3, 5, 7, 11, 13, 17, and 19. The phis of these were: 2 = 1 3 = 2 5 = 4 7 = 6 11 = 10 13 = 12 17 = 16 From this it can be seen that the phi of a prime number is always one less than the number in question e.g. 17 = 16. This is because there a no factors which go into a prime number, except for the number 1 and itself, and in the phi function the number in question is not considered, and the number 1 is always a factor, therefore all numbers except for the number in question, are all counted. We can now draw from this a simple formula where 'n' is any given PRIME number: n = n-1 Odd or Even? All of the results were even, except for the phi of 2 which is 1. Factors within a Phi Other than Prime numbers themselves and numbers which have factors which are prime numbers, have a phi of how many prime numbers exist in them, such as the phi of 8: 8 = 1, 3, 5, 7 In this phi only the numbers: 1, 3, 5 and 7 do not go into the number 8, and these numbers are all prime, except for 2 which is a factor of all even numbers therefore the phi of 8 has the same number of prime numbers in it, as the value of its phi, which is 4. This pattern also exists in the phis of: 3 4 8 already stated 10 14 16 The Phi's of Powers of Prime Number How to work out the Phi's of Powers of Prime Number The phi of a power of a prime number can still be found by identifying the number of positive integers less than n, which have no factor other than 1 in common are co-prime with n, but a simple amount of BIDMASS must be employed. This means that the phi of a power of a prime number can be found but multiplying the brackets out first, must occur, and then the phi of that number can be found. Below the first 3 prime numbers in the sequence of prime numbers 2, 3, 5, 7"¦.. and there first 5 power numbers and their phis have been found. Powers of Prime Numbers and their Phi's ** In this next part of my study the sign ^ will indicate a powered number, such as 2^2, 2^3, 2^4"¦"¦.. ** 2 = 1 2^2 = 2 2^3 = 4 2^4 = 8 2^5 = 16 3 = 2 3^2 = 6 3^3 = 18 3^4 = 54 3^5 = 162 5 = 4 5^2 = 20 5^3 = 100 5^4 = 500 5^5 = 2500 From the prime numbers and their powers phi's it can be seen that a clear pattern has become apparent. It can be seen that each time the power goes up by 1, the phi of that powered prime, is the previous powered prime's phi, multiplied by the prime number in question. I know this because it is evident in the previously stated phi's, for example from the powers of 2 phi functions, 1 x 2 is 2, 2 x 2 is 4, 4 x 2 is 8 and so on.   

Introduction In this investigation I have been given the task of discovering a formula to tell us the value of n when 'n' is any given number. Also, in the second part of my investigation, look into whether there is a link between the phi of a number,...

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