Suppose that p and q are distinct primes. Use the principle...

Suppose that $p$ and $q$ are distinct primes. Use the principle of inclusion-exclusion to find $\phi(p q),$ the number of positive integers not exceeding $p q$ that are relatively prime to $p q$ .

Key Concepts

Relatively Prime

Two integers are said to be relatively prime (or coprime) if they share no prime factors, meaning their greatest common divisor (gcd) is 1. This concept is fundamental in number theory, particularly in defining Euler’s totient function, where the focus is on counting the integers that do not share any common factor with a given number.

Euler's Totient Function

This function, typically denoted as ?(n), counts the number of positive integers up to n that are relatively prime to n. It plays a central role in number theory and has applications in topics such as modular arithmetic and cryptography. The function is multiplicative and for any number that is a product of distinct primes, its value can be computed using the formula ?(n) = n × ?(1 - 1/p), where the product is over the distinct prime factors of n.

Prime Numbers

Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves. They are the basic building blocks of the natural numbers, as every integer greater than 1 can be uniquely factored into prime numbers. In problems involving arithmetic functions like Euler’s totient function, the properties of primes simplify calculations and lead to clear factorization.

Principle of Inclusion-Exclusion

This combinatorial principle is used to count the number of elements in the union of overlapping sets by including the sizes of individual sets and excluding the sizes of their intersections, with adjustments for further overlaps. In the context of counting numbers relatively prime to a composite number, it is applied to subtract those numbers divisible by each prime factor and to correct for overcounting by adding back in the numbers counted more than once.

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