GCSE: Phi Function
Meet our team of inspirational teachers

GCSE Maths questions
 Develop your confidence and skills in GCSE Maths using our free interactive questions with teacher feedback to guide you at every stage.
 Level: GCSE
 Questions: 75

Beyond Pythagoras
When I put the numbers into the table like this, I realised two things, firstly that 'a' increases by two, and secondly that 'c' is b+1 on both of the terms. As I currently only have two numbers in the sequence for 'a', my assumption about 'a' increasing by two each time is not necessarily correct, because the sequence's 1st difference could go up by two instead of the difference staying the same each time. To check that my prediction is correct, I must come up with a formula to find 'b' from 'a', and therefore be able to find 'c'.
 Word count: 2083

a. Describe Aristotle's teaching about the difference between the Final Cause and other sorts of causes.
These substances create the form of an object; Aristotle also questioned what causes these objects to have the characteristics that it portrays? If a chair had only three legs, would it still be a chair? Aristotle concluded that these questions can be answered in four different ways or four different causes, this was the best way explain why things are the way they are. The Material cause answers the question for what things are made of, but this was only the first cause meaning that it is not enough on its own.
 Word count: 684

Identify and explain the rules and equations associated with the Phi function.
21, 22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38 24 ?40 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39 16 ?41 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40 40 ?42 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41 12 ?43 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42 42 ?44 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43 20 ?45 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44 24 ?46 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44,45 22 ?47 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44,45,46 46 ?48 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44,45,46,47 16 ?49 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44,45,46,47,48 42 ?50 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44,45,46,47,48,49 20 ?51 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40, 41,42,43,44,45,46,47,48,49,50 32 ?52 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40, 41,42,43,44,45,46,47,48,49,50,51 24 ?53 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40, 41,42,43,44,45,46,47,48,49,50,51,52 52 ?54 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40, 41,42,43,44,45,46,47,48,49,50,51,52,53 18 ?55 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40, 41,42,43,44,45,46,47,48,49,50,51,52,53,54 40 In my coursework there are some examples which range up to 60 these were obtained from a reliable website and checked over to ensure they were correct.
 Word count: 3837

Binary Integers
= 148 (denary) So if you want to convert a binary integer into its denary equivalent, just write it in a table like this and add up the values in the columns which have a bit value of 1. Convert binary integers to denary Work out which denary integers are represented by these Binary Integers.
 Word count: 526

Investigating the Phi function
= 8 (21) 3,6,9,12,15,18,7,14 1,2,4,5,8,10,11,13,16,17,19,20 (21) = 12 (22) 2,4,6,8,10,12,14,16,18,20,11 1,3,5,7,9,13,15,17,19,21 (22) = 10 (23) 3,6,12,8,15,18,21,16 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 16,17,18,19,20,21 (23) = 22 (24) 2,4,6,8,10,12,14,16,18,20,22 6,3 1,5,7,11,13,17,19,23 (24) = 8 Part 1 i. (3) = 2 ii. (8) = 4 iii. (11) = 10 iv. (24) = 8 Trends I spotted: * from my observations I noticed simple trends like: (2) = 1 Answer is half the phi number When you double the phi number the answer for that number is double the original answer (also these phi numbers are powers of 2) (4) = 2 (8) = 4 (16) = 8 (32)
 Word count: 2161

The Phi Function
Therefore this shows that the phi function of 3 is 2. (2) ?(8) = 4 The factors of 8 are: 1, 2, 4, and 8. The integers, which are less than 8, are 1, 2, 3, 4, 5, 6 and 7. The table below is similar to the one above, but is for the phi function of 8. Integers Factors Does it fit into expression? Yes or No 1 1 yes 2 1,2 No 3 1,3 Yes 4 1,2,4 No 5 1,5 Yes 6 1,2,3,6 No 7 1,7 Yes From looking at the above table I can tell you that the phi function of 8 is 4.
 Word count: 1955

Investigate the strength of a snail's mucus on different surfaces
 Oil (to see what affect it has on the mucus)  Water (to see what affect it has on the mucus)  Ruler  Protractor  Snail Method 1. Weight the snails and label each one (mass in grams) 2. Put snail number 1 on surface A at 10 cm away for the central point and wait for a minute or two for the snail to stick to it. 3. Hold the surface at different angles for 30 seconds at a time then move angle.
 Word count: 1379

The totient function.
This is a good way of checking if your Phi answer is wrong or not. Even for even numbers the Phi value increases but this is not applicable to the even numbers that are the factors of 3 and 5. The Phi value also increases for odd numbers as the number increases but this does not include odd numbers. 2) Another common sequence that I found was that for all the Phi values of the prime numbers, the Phi value was one less than that number.
 Word count: 3982

Millikan's theory.
Examining these two aspects in detail will enable us to better grasp Millikan's theory of representation. NORMAL/PROPER FUNCTIONS Millikan's notion of Normal and proper functions lies at the root of her conception of representation. The capitalization of "Normal" here is meant to distinguish the concept from causal or dispositional senses of the term  for indeed, Millikan's concept of proper or Normal functions is a normative one. The Normal function of a given item specifies what that item should do in terms of its evolutionary history, what it has been evolutively designed to do (which may or may not be what it actually does), not the events it causes or tends to cause.
 Word count: 3066

The phi function.
(24) = 8 This is because there are 8 positive integers less than 24 which have no common factors with 24 other than 1. B) Obtain the Phi Function for at least 5 positive integers of your own choice. I intend to find the Phi Function of the following: 1 ? (5) because it is a prime number. 2 ? (10) because its an even number. 3 ? (15) because its an odd number 4 ? (16) because it's squared 5 ?
 Word count: 2134

Describe Aristotle's teachings about the differences between the final cause and the other sorts of causes.
Aristotle believed that not only is material cause just the substance something is made from, but also the means by which a thing is brought about (how we see it and can tell what it is). The efficient cause is the means in which something actually becomes something, which includes the maker, the tools and the skills being used. The efficient cause along with all the causes play an important part in causality. Without the efficient cause the skills/maker/tools would not be present, which means that the final cause would not be produced.
 Word count: 1046

The Phi function.
But as we see the phi value for 7 and 9 is the same and so we cannot take this observation into consideration. Also we cannot say that as n increases the value of ?(n) also increases. There is another observation however. The phi value for all odd numbers that are not prime increases by multiples of 2. For example the ?(9)= 6 and then the ?(15)=8 and the ?(21)=12 and ?(25)=20. If we arrange all these results as 6,8,12,20 we will see that the difference between them is 2,4,8.
 Word count: 3155

The PhiFunction.
Other than the process of elimination, one may find the phi function of a positive integer by seeking the coprimes of n directly. Taking the same example, ?(6) we may say from the very beginning, without needing too many steps, that 1 and 5 are coprime with 6 therefore ?(6) = 2, taking another example, we may say that1,2,3,4,5,6 are all coprime with 7 therefore ?(7)=6. 1a) I have devised a table to conveniently find out ?(n) n 3 8 11 24 0 ?
 Word count: 4988

In this coursework I was asked to investigate the Phi Function (f) of a number (n).
I will create a table for the Phi values of the numbers from 1 to 30. I have created this table because it would be easier to lookup the phi values of the numbers within this range instead of solving it every time I need to find a value quickly. In this table I have crossed out the factors that are not coprime and left the coprime ones. In the end of each row, I counted the uncrossed numbers to find the phi value of that number. ?(2); 1, = 1 ?(3); 1 2; =2 ?(4); 1 2 3; =2 ?(5); 1 2 3 4; =4 ?
 Word count: 2723

The phifunction
Yet what would happen of you needed to know the phi of 30,041. To complete my goal I will have to follow many different stages to finally work out the formula. So the first stage I will do is to find out the numbers from say 1 to 36. n The ?(n) The numbers with no common factors with n. 1 0 0. 2 1 1. 3 2 1.2 4 2 1,3. 5 4 14. 6 2 1,5. 7 6 16.
 Word count: 1719

Mathematics GCSE Coursework  The Phi Function.
x ?(4)=6 x 2=12; So ?(7 x 4) = ?(7) x (4) is true. 2) ?(6 x 4)=?(6) x ?(4); ?(6)=2; 1 2 3 4 5 6 ?(4)=2; I have found ?(4) already. ?(6 x 4)= ?(24)=8; I have found ?(24) already. ?(6) x ?(4)=2 x 2=4; So ?(6 x 4)=?(6) x ?(4) is true. b) ?(5 x 2)=?(5) x ?(2); ?(5)=4; I have found ?(5) already. ?(2)=1; 1 2 ?(5 x 2)= ?(10)=4; 1 2 3 4 5 6 7 8 9 10 ?(5) x ?(2)=4 x 1=4; So ?(5 x 2)=?(5) x ?(2) is true. ?(7 x 3)=?(7)
 Word count: 1324

Maths Primes and Multiples Investigation
Having done these examples I below have drawn up a table of my results. When I made the table I saw that there are lots of other combinations that I need to investigate. Prime Non Prime Odd Even Multiple Prime Yes Yes Non Prime Odd Even No Multiple No 3. To complete my table and to test out this idea I will try a variety of numbers in different combinations. For example I will be using odds, evens and prime numbers in different combinations.
 Word count: 3020

The Phi Function Investigation
I am using 17 as it is a prime number.... 1, 2, 3 , 4, 5, 6 , 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 = 16 iii) o(15) I am using 15 as it is a odd number.... 1, 2, 3 , 4, 5, 6 , 7, 8, 9, 10, 11, 12, 13, 14, 15 = 8 iv) o(16) I am using 16 as it is square number.... 1, 2, 3 , 4, 5, 6 , 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 = 8 v)
 Word count: 1394

The Phi Function Investigation
Obtain the PhiFunction for at least 5 positive integers of your own choice. (I) ?(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. (ii) ?(10): 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 10 = 1,3,7,9 The number 10 has 4 positive integers, they are shown above.
 Word count: 2358