What is the value of phi(100) in the Euler’s phi function?

Answer 1

Euler's totient function (φ) can be thought of counting the positive integers up to a given integer n that are relatively prime to n

As it turns out...

φ(n) = φ((p₁)^k1) x φ((p₂)^k2) x ... x φ((p_m)^km)

Where the p_m are the prime numbers used to represent n. k_m are the number of factors of each prime number representation.

So φ(100) = φ(10²) = φ(2²)φ(5²) = n x ( 1 - 1/p₁) * (1 - 1/p₂ ) = 100 * (1 - 1/2) * (1 - 1/5) = 40

But φ((p₁)^k1) can be represented as p₁^k1 x (1 - 1/p₁)

So the terms in the multiplication can be written as -->

φ(2²) = 2² * (1 - 1/2) = 4*(1/2) = 2

φ(5²) = 5² * (1 - 1/5) = 25*(4/5) = 20

So φ(100) = φ(10²) = φ(2²)φ(5²) = 2 * 20 = 40

Also --> φ(n) = φ((p₁)^k1) x φ((p₂)^k2) x ... x φ((p_m)^km) = n x ( 1 - 1/p₁) * (1 - 1/p₂ ) x ... x (1 - 1/p_m)