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.
(3) φ(11) =10
The factors of 11 are: 1 and 11.
The integers, which are less than 11, are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. The table below is similar to the one above, but is for the phi function of 11.
By looking at the table above, I can tell you that the phi function of 11 is 10 since all integers fit into the expression.
(4) φ(24) = 8
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.
The integers, which are less than 24, are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 and 23. The table below is similar to the one above, but is for the phi function of 24.
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.
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. The table below shows integers which are below 4, 7 and 28 with their factors. It also contains whether it fits into the expression and which number (4, 7 and 28) it applies to.
From the above table you can see that φ(28) = 12, also that φ(7) = 6 and φ(4) = 2. This shows that phi 28 is equal to phi 7 times phi 4 because the phi function of 7, which is 6 multiplied by the phi function of 4, which is 2 gives you the phi function of 28, which is 12.
(2) φ(6x4) = φ(24) = 8, φ(6) = 2 and φ(4) = 2
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24. The factors of 6 are: 1, 2, 3 and 6. The factors of 4 are: 1, 2 and 4.
The integers, which are less than 24, are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 and 23. The integers below 6 are: 1, 2, 3, 4 and 5. The integers which are below 4 are: 1, 2 and 3. The table below shows integers which are below 4, 6 and 24 with their factors. It also contains whether it fits into the expression and which number (4, 6 and 24) it applies to.
From the above table you can see that φ(24) = 8, φ(6) = 2 and φ(4) = 2. This shows that phi 24 is not equal to phi 6 times phi 4 because the phi function of 6, which is 2 multiplied by the phi function of 4, which is 2 does not equal to the phi function of 24, which is 8, but instead it equals to 4.
(3) φ(4x3) = φ(12) = 4, φ(4) = 2, φ(3) = 2.
The factors of 12 are: 1, 2, 3, 4, 6 and 12. The factors of 4 are 1, 2 and 4. The factors of 3 are: 1 and 2.
The integers which are less than 12 are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. The integers which are less than 4 are 1, 2 and 3. The integers which are less than 3 are: 1 and 2. The table below shows integers which are below 3, 4 and 12 with their factors. It also contains whether it fits into the expression and which number (3, 4 and 12) it applies to.
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.
