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: f(3), f(8), f(11) and f(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: f(7x4) = f(7) x f(4) and f(6x4) =
f(6) x f(4) then create some of my own and check that if they are
equal to each other or not.
Part 1
======
(1) f(3) = 2
The factors of f(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
This table shows you that the number of positive integers less than
three and has no other common factor other than 1 is two integers: 1
and 2. Therefore this shows that the phi function of 3 is 2.
(2) f(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