Euler's φ (phi) Function counts the number of positive integers not exceeding n and relatively prime to n. Traditionally, the proof involves proving the φ function is multiplicative and then proceeding to show how the formula arises from this fact. We ignore this fact, at least directly, and show a practical and sound method to calculate φ. We offer a proof of the closed form formula for this function relying on similar, but subtly different counting techniques.
Chorge, Shashank and Vargas, Juan
"Proof of Euler's φ (Phi) Function Formula,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 14:
2, Article 6.
Available at: https://scholar.rose-hulman.edu/rhumj/vol14/iss2/6