Calculating Euler’s Totient Function ϕ(92)

Euler’s Totient function, denoted by ϕ($n$), is a fundamental concept in number theory. It counts the positive integers up to a given integer $n$ that are relatively prime to $n$. Two integers are relatively prime if their greatest common divisor (GCD) is 1.

To calculate ϕ($n$), we first need to find the prime factorization of $n$. If the prime factorization of $n$ is given by $n = p_1^{k_1} p_2^{k_2} \dots p_r^{k_r}$, where $p_1, p_2, \dots, p_r$ are distinct prime factors and $k_1, k_2, \dots, k_r$ are their respective positive integer exponents, the formula for Euler’s Totient function is:

$\varphi(n) = n \left(1 - \frac{1}{p_1}\right) \left(1 - \frac{1}{p_2}\right) \dots \left(1 - \frac{1}{p_r}\right)$

Alternatively, the formula can be written as:

$\varphi(n) = p_1^{k_1-1}(p_1-1) p_2^{k_2-1}(p_2-1) \dots p_r^{k_r-1}(p_r-1)$

Step-by-Step Calculation of ϕ(92)

Let's find the value of ϕ(92) using these steps:

1. Find the prime factorization of 92:

   We divide 92 by the smallest prime numbers:

   • $92 \div 2 = 46$
   • $46 \div 2 = 23$
   • 23 is a prime number.

   So, the prime factorization of 92 is $2 \times 2 \times 23 = 2^2 \times 23^1$.

   The distinct prime factors are $p_1 = 2$ and $p_2 = 23$. The exponents are $k_1 = 2$ and $k_2 = 1$.

2. Apply the Euler’s Totient function formula:

   Using the formula $\varphi(n) = n \prod (1 - \frac{1}{p_i})$:

   $\varphi(92) = 92 \left(1 - \frac{1}{2}\right) \left(1 - \frac{1}{23}\right)$

   $\varphi(92) = 92 \left(\frac{2-1}{2}\right) \left(\frac{23-1}{23}\right)$

   $\varphi(92) = 92 \left(\frac{1}{2}\right) \left(\frac{22}{23}\right)$

   $\varphi(92) = \frac{92 \times 1 \times 22}{2 \times 23}$

   $\varphi(92) = \frac{92 \times 22}{46}$

   Since $92 = 2 \times 46$, we can simplify:

   $\varphi(92) = \frac{(2 \times 46) \times 22}{46}$

   $\varphi(92) = 2 \times 22$

   $\varphi(92) = 44$

   Alternatively, using the formula $\varphi(n) = p_1^{k_1-1}(p_1-1) p_2^{k_2-1}(p_2-1) \dots$:

   $\varphi(92) = \varphi(2^2 \times 23^1)$

   $\varphi(92) = (2^{2-1}(2-1)) \times (23^{1-1}(23-1))$

   $\varphi(92) = (2^1(1)) \times (23^0(22))$

   $\varphi(92) = (2 \times 1) \times (1 \times 22)$

   $\varphi(92) = 2 \times 22$

   $\varphi(92) = 44$

Both methods yield the same result. The value of Euler’s Totient function ϕ(92) is 44.

Confirmation with Options

Let's look at the given options:

Our calculated value, 44, matches Option 1.

Revision Table: Euler's Totient Function Basics

Additional Information on Euler's Totient Function Properties

Euler's Totient function has several interesting properties:

• It is a multiplicative function: If $m$ and $n$ are relatively prime, then ϕ($mn$) = ϕ($m$)ϕ($n$). For example, $92 = 4 \times 23$, and $\text{GCD}(4, 23) = 1$. $\varphi(4) = \varphi(2^2) = 2^{2-1}(2-1) = 2(1) = 2$. $\varphi(23) = 23-1 = 22$. So, $\varphi(92) = \varphi(4 \times 23) = \varphi(4) \times \varphi(23) = 2 \times 22 = 44$. This confirms our calculation.
• The sum of the values of ϕ($d$) for all positive divisors $d$ of $n$ is equal to $n$. That is, $\sum_{d|n} \varphi(d) = n$. For $n=92$, the divisors are 1, 2, 4, 23, 46, 92.
  • $\varphi(1) = 1$
  • $\varphi(2) = 1$
  • $\varphi(4) = \varphi(2^2) = 2$
  • $\varphi(23) = 22$
  • $\varphi(46) = \varphi(2 \times 23) = \varphi(2) \times \varphi(23) = 1 \times 22 = 22$
  • $\varphi(92) = 44$

  Sum = $1 + 1 + 2 + 22 + 22 + 44 = 92$. This property holds true for 92.

• Euler's theorem states that if $a$ and $n$ are relatively prime positive integers, then $a^{\varphi(n)} \equiv 1 \pmod{n}$. This theorem is a generalization of Fermat's Little Theorem.