Question

Show that if \(a\) and \(n\) are relatively prime with \(n>0\), then ord \(_{n} a \mid \phi(n)\).

Short answer

Question: Prove that ord$_{n}a | \phi(n)$ for relatively prime numbers $a$ and $n$. Answer: We can prove this result using properties of modular arithmetic and Euler's theorem. By definition, ord$_{n}a$ is the smallest positive integer $k$ such that $a^k \equiv 1 \pmod{n}$. Being relatively prime, Euler's theorem applies and we have $a^{\phi(n)} \equiv 1 \pmod{n}$. Let $d = $ ord$_{n}a$. Using the division algorithm, we can write $\phi(n) = pd + r$, where $0 \le r < d$. It can be shown that $a^r \equiv a^{\phi(n)} \pmod{n}$, leading to the conclusion that $r = 0$. This implies $\phi(n) = pd$, which means ord$_{n} a = d \mid \phi(n)$.

Step-by-step solution

Definition of the order

The order of \(a\) modulo \(n\), denoted ord \(_{n} a\), is the smallest positive integer \(k\) such that \(a^k \equiv 1 \pmod{n}\).

Prove that \(a^{\phi(n)} \equiv 1 \pmod{n}\)

Since \(a\) and \(n\) are relatively prime, Euler's theorem applies and we have \(a^{\phi(n)} \equiv 1 \pmod{n}\).

Use the order property and the divisibility statement

Let \(d =\) ord \(_{n} a\). By definition of the order, we have \(a^d \equiv 1 \pmod{n}\), and \(d\) is the smallest positive integer with this property. Since \(a^{\phi(n)} \equiv 1 \pmod{n}\) (from Step 2), by the division algorithm, we can write \(\phi(n) = pd + r\), where \(p\) and \(r\) are integers such that \(0 \le r < d\).

Analyze the remainder term

Now consider \(a^r \equiv a^{\phi(n) - pd} \pmod{n}\). Using properties of modular exponentiation, we have \(a^{\phi(n)} \equiv a^{pd + r} \equiv a^{pd}a^r \equiv (a^d)^pa^r \equiv 1^pa^r \equiv a^r \pmod{n}\). By definition of the order, since \(0 \le r < d\), we can conclude that \(r = 0\).

Show that ord \(_{n} a \mid \phi(n)\)

Since \(r = 0\), we have \(\phi(n) = pd\), which means ord \(_{n} a = d \mid \phi(n)\). This completes the proof.

Key concepts

Order Modulo n

In the realm of number theory, the concept of order modulo n plays a pivotal role. It is defined for an integer a with respect to a modulus n. Specifically, if a and n are integers where n > 0 and a is not a multiple of n, the order of a modulo n, denoted as ord_n a, is the smallest positive integer k such that a^k \(equiv 1 \pmod{n}\).

Understanding the order modulo n is essential when delving into concepts like cyclic groups and powers in modular arithmetic. In essence, it measures how frequently a given number's powers will 'cycle back' to 1 when considered under a specified modulus. The existence of such an order for every a that is coprime to n (relatively prime, as we'll see below) is a foundation stone upon which many results in number theory are built, including the famous Euler's theorem.

Totient Function

The totient function, often denoted by \(\phi(n)\), is a function that counts the number of positive integers up to a given integer n that are relatively prime to n. This function, also referred to as Euler's totient function, plays an integral role in Euler's theorem and in the RSA encryption algorithm - a cornerstone of modern cryptography.

To compute the totient of a number, one can use various methods depending on whether the prime factors of the number are known. For instance, if a number n is prime, then \(\phi(n) = n - 1\) because all the numbers less than n are relatively prime to it. For a number with prime factors, there's a formula involving the product of the factors raised to their power minus one and then reduced by the same factor. The totient function is multiplicative, meaning that if two numbers a and b are coprime, then \(\phi(ab) = \phi(a)\cdot\phi(b)\).

Modular Exponentiation

The process of modular exponentiation is a method by which one computes the remainder when an integer raised to a given power is divided by a modulus. Expressed mathematically, this means finding \(a^b \mod n\) for given integers a, b, and n. We often deal with very large integers in applications such as cryptography, so fast algorithms like the square-and-multiply method are used to compute modular exponentiation efficiently.

Understanding modular exponentiation is crucial when studying properties related to Euler's theorem or solving modular equations. Notably, modular exponentiation respects many of the regular properties of exponentiation but must be treated carefully, for instance, when working with the previously mentioned order modulo n. Because of modular arithmetic's 'wrap-around' nature, this operation enables us to handle astronomic numbers practically.

Relatively Prime

Two integers are said to be relatively prime (or coprime) if they share no common positive divisors except for 1. Essentially, their greatest common divisor (GCD) is 1. For example, 8 and 15 are relatively prime, while 12 and 18 are not because they are both divisible by 6 (among other numbers).

This concept is a linchpin in various theorems and algorithms in number theory and is particularly significant when discussing modular arithmetic and order modulo n. A key point in understanding relatively prime numbers is their ubiquity in conditions for the application of many important theorems, such as Euler's theorem, which necessitates the two numbers involved to be relatively prime for the theorem to hold true. As it turns out, the relationship between numbers being relatively prime and properties of modular exponentiation is a deep and powerful tool that resounds throughout mathematics and cryptography.