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.

Author Bio

Shashank Chorge was born in India. He is a computer engineer from Bombay University. He teaches mathematics for aptitude exams and likes to solve puzzles and problems especially of Olympiads. He is interested in Number theory and algebra. In past time he likes to watch cricket and movies. Faculty Sponsor:

Juan Vargas was born in Dominican Republic. He is a senior mathematics major studying at UNC Charlotte who is currently in the MASS Program at Pennsylvania State University. He plays chess and Go online when time permits. In general he likes to play tennis in summer time, study summations to obtain exact values and learn about interesting problems in Number Theory and Group Theory.