Each of exercises 51-53 refers to the Euler phi function, denoted , which is defined as follows: For each integer is the number of positive integers less than or equal to n that have no common factors with n except . For example, because there are four positive integers less than or equal to 10 that have no common factor with 10 except -namely, 1,3,7, and9.
Prove that if p is a prime number and n is an integer with , then .
When is a prime number and is an integer with , then
is a prime number and is an integer with
There are integers between divisible by .
The positive integers less than and co-prime to are different from the divisors of .
are the divisors of . These are in number
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