Question

How many elements have multiplicative inverses in \(Z_{p q}\) when \(p\) and \(q\) are primes?

Short answer

There are \((p-1)(q-1)\) elements with multiplicative inverses in \(Z_{pq}\).

Step-by-step solution

Understand the Structure of \(Z_{pq}\)

The set \(Z_{pq}\) consists of all integers from 0 to \(pq - 1\), where any element \(x\) that is coprime to the modulus \(pq\) has a multiplicative inverse. Here, \(p\) and \(q\) are distinct primes.

Identify the Elements that Have Inverses

A number has a multiplicative inverse modulo \(n\) if it is coprime to \(n\). An element \(a\) is coprime to \(n\) if \(\gcd(a, n) = 1\). Since \(n = pq\), we look for numbers that are not multiples of either \(p\) or \(q\).

Calculate Euler’s Totient Function

Euler's totient function \(\varphi(n)\) counts the number of integers up to \(n\) that are coprime to \(n\). For \(n = pq\), where \(p\) and \(q\) are primes, \(\varphi(pq) = (p-1)(q-1)\).

Compute \(\varphi(pq)\)

Substituting into Euler’s formula, we calculate \(\varphi(pq) = (p - 1)(q - 1)\). This accounts for all numbers less than \(pq\) that are not divisible by \(p\) or \(q\).

Conclusion

The total number of elements in \(Z_{pq}\) that have multiplicative inverses is \(\varphi(pq) = (p-1)(q-1)\).

Key concepts

Euler’s Totient Function

Euler's Totient Function, often denoted as \(\varphi(n)\), is a fascinating and crucial concept in number theory. It is used to count how many integers up to \(n\) are coprime with \(n\). This particular function helps in identifying the integers that have multiplicative inverses in modular arithmetic.

Consider \(n = pq\), where \(p\) and \(q\) are distinct primes. The formula for Euler's Totient Function in this case is given by:

• \(\varphi(pq) = (p-1)(q-1)\)

This expression derives from the fact that there are \(p - 1\) numbers less than \(p\) that are coprime to \(p\), and \(q - 1\) numbers less than \(q\) that are coprime to \(q\). Together, excluding the multiples of \(p\) and \(q\) from the range up to \(pq\), we find the total using the product formula.

Therefore, Euler’s Totient Function helps calculate how many numbers in \(Z_{pq}\) have multiplicative inverses.

Coprime Numbers

Coprime numbers, or relatively prime numbers, are two integers that have no common positive divisors other than 1. This means that the greatest common divisor (\(\gcd\)) of these numbers is 1. Understanding coprime numbers is crucial when working with multiplicative inverses in modular arithmetic.

If we consider an integer \(a\) and a modulus \(n\), \(a\) has a multiplicative inverse if and only if \(a\) is coprime to \(n\). For example, in the case of \(Z_{pq}\) where \(p\) and \(q\) are distinct primes, an integer \(x\) is coprime to \(pq\) if it is neither a multiple of \(p\) nor a multiple of \(q\).

Here are some key points on coprime numbers:

• Coprimeness can be checked using the \(\gcd\).
• If \(\gcd(a, n) = 1\), then \(a\) and \(n\) are coprime.

Understanding which numbers are coprime with a given modulus helps in determining the elements with multiplicative inverses.

Greatest Common Divisor

The Greatest Common Divisor (\(\gcd\)) of two integers is the largest positive integer that divides both numbers without leaving a remainder. Calculating the gcd is one of the simplest yet most effective methods to determine if two numbers are coprime.

In modular arithmetic, finding the \(\gcd\) helps establish whether a number has a multiplicative inverse. If the \(\gcd\) of an integer \(a\) and a modulus \(n\) is 1, then \(a\) and \(n\) are coprime, leading \(a\) to have an inverse.

This simple approach plays a powerful role when dealing with systems such as \(Z_{pq}\). By ensuring that numbers are coprime to the modulus, we are able to identify those that possess inverses. Here's a brief run-through:

• The gcd can be calculated using the Euclidean algorithm.
• If \(x\) divides both \(a\) and \(n\) without remainder other than \(1\), then \(x = 1\).

With these calculations, understanding the relationships between numbers in modular systems becomes more manageable and precise.