Euler's totient function (also know as Euler's phi function and denoted with the Greek letter φ) counts the positive integers up to a given integer n that are relatively prime to n. More formally, it counts the number of integers k in the range 1≤k≤n for which: gcd(n, k)=1.  The following are rules can be used for fast computation of phi, φ(j): if n is prime, then φ(n)=n-1   -- same as Fermat’s Little Theorem If gcd(n,m)=1, then φ(n*m)=φ(n)*φ(m) if p is prime, then φ(pn)=pn-pn-1.  Employing the rules above what is φ(17)?单项选择题

A

16

B

7

C

1

D

8

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Euler's totient function (also know as Euler's phi function and denoted with the Greek letter φ) counts the positive integers up to a given integer n that are relatively prime to n. More formally, it counts the number of integers k in the range 1≤k≤n for which: gcd(n, k)=1.  The following are rules can be used for fast computation of phi, φ(j): if n is prime, then φ(n)=n-1   -- same as Fermat’s Little Theorem If gcd(n,m)=1, then φ(n*m)=φ(n)*φ(m) if p is prime, then φ(pn)=pn-pn-1.  Employing the rules above what is φ(35)?

Consider Euclid's Proof (By Refutation) that there are infinitely many primes : Proof: Suppose that there were only finitely many prime numbers. Then, we could write them all down in a list as, {p1,p2,...pn}. Consider now their product plus one, giving the new number:  P = (p1 x p2 x.. x pn)+1 P is necessarily larger than any of the primes on our list.  What remainder does this new number P leave when divided by any one of the pi for 1 ≤ i ≤ n?

Suppose p is prime and, a and b are natural numbers, and  . Then either or .

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