In the particular case when m is prime say p, Euler’s theorem turns into the so-called Fermat’s little theorem : a p-1 ≡ 1 (mod p)ħ) Number of generators of a finite cyclic group under modulo n addition is Φ(n). The RSA cryptosystem is based on this theorem:

(or relatively prime) positive integers, then Proof :, where a and b are prime numbers, \Total number from 1 to ab = ab Total multiples of a from 1 to ab = Total multiples of b from 1 to ab = Example:a = 5, b = 7, ab = 35Multiples of a = 7 n = 1 + 1 + 4 + 4 = 10Ħ) The most famous and important feature is expressed in Euler’s theorem : The theorem states that if n and a are coprime Set: result = result * (1.0 - (1.0 / (float) p)) īelow is the implementation of Euler’s product formula. 1) Initialize : result = nĢ) Run a loop from 'p' = 2 to sqrt(n), do following for every 'p'. We can find all prime factors using the idea used in this post. For example value of Φ(6) = 6 * (1-1/2) * (1 – 1/3) = 2. This function is usually called the Euler function or the Euler totient function and has many connections to number theory. The formula basically says that the value of Φ(n) is equal to n multiplied by-product of (1 – 1/p) for all prime factors p of n. The idea is based on Euler’s product formula which states that the value of totient functions is below the product overall prime factors p of n. Therefore, an upper bound on the time complexity of the above solution is O(N log N) īelow is a Better Solution. The time complexity of the gcd function is O(h) where “h” is the number of digits in a smaller number of given two numbers. The above code calls gcd function O(n) times. ISRO CS Syllabus for Scientist/Engineer Exam.ISRO CS Original Papers and Official Keys.GATE CS Original Papers and Official Keys.

Graph of euler totient function