Thus, $\phi(48) = \boxed{16}$.

["# Thus, $\phi(48) = \boxed{16}$: The Euler’s Totient Function Explained", "Mathematics is full of fascinating sequences and properties that underpin number theory, cryptography, and beyond. One such powerful concept is Euler’s Totient Function, denoted $\phi(n)$, which counts how many positive integers less than or equal to $n$ are relatively prime to $n$. Understanding how to compute $\phi(n)$ is essential, and in this article, we explore why $\phi(48) = 16$ using clear reasoning and step-by-step calculation.", "## What Is Euler’s Totient Function?", "Euler’s Totient Function, $\phi(n)$, is defined as the number of integers between $1$ and $n$ (inclusive) that share no common factors with $n$ other than $1$. In other words, $\phi(n)$ measures how many numbers from $1$ to $n$ are coprime to $n$. For example, $\phi(6) = 2$ because only $1$ and $5$ are coprime to $6$.", "The function has deep applications in number theory and is key to RSA encryption, where it helps determine valid keys. A fundamental recursive property arises from prime factorization: if $n = p_1^{k_1} p_2^{k_2} \cdots p_m^{k_m}$, then\n$$\n\phi(n) = n \left(1 - \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right) \cdots \left(1 - \frac{1}{p_m}\right).\n$$", "## Step-by-Step Calculation: Why $\phi(48) = 16$", "To compute $\phi(48)$, we begin by factoring $48$:\n$$\n48 = 2^4 \ imes 3^1.\n$$", "Using the multiplicative property of $\phi$,\n$$\n\phi(48) = \phi(2^4) \ imes \phi(3^1).\n$$", "We calculate each totient separately:", "### Step 1: Compute $\phi(2^4 = 16)$", "Applying the formula for prime powers:\n$$\n\phi(2^4) = 2^4 \left(1 - \frac{1}{2}\right) = 16 \ imes \frac{1}{2} = 8.\n$$", "These are the odd numbers from $1$ to $15$: $1, 3, 5, 7, 9, 11, 13, 15$ — totaling $8$ numbers.", "### Step 2: Compute $\phi(3^1 = 3)$", "Similarly,\n$$\n\phi(3) = 3 \left(1 - \frac{1}{3}\right) = 3 \ imes \frac{2}{3} = 2.\n$$", "These numbers are $1$ and $2$.", "### Step 3: Multiply totients", "Now multiply the results:\n$$\n\phi(48) = \phi(16) \ imes \phi(3) = 8 \ imes 2 = 16.\n$$", "Thus, $\phi(48) = 16$ confirms that exactly $16$ integers between $1$ and $48$ are coprime to $48$.", "## Why Is This Important?", "Understanding $\phi(48) = 16$ reveals more than just a count—it illustrates how prime factorization simplifies complex counting. This principle extends to algorithms in computer science, primality testing, and encryption protocols like RSA, where totients are used to derive modular inverses and secure key exchanges.", "Additionally, numbers coprime to $48$ are precisely those that can serve in multiplicative operations modulo $48$, forming the multiplicative group of integers modulo $n$. This group has order $\phi(48) = 16$, underscoring $\phi(48)$’s role in modular arithmetic structure.", "## Conclusion", "Euler’s Totient Function, $\phi(n)$, elegantly counts coprime integers and plays a pivotal role in both theoretical and applied mathematics. The computation $\phi(48) = 16$ exemplifies how factoring and prime decomposition enable efficient evaluation. Whether用于 cryptography, algorithm design, or pure number theory, mastering $\phi(n)$ deepens mathematical insight and strengthens problem-solving capabilities.", "So the boxed answer stands:\n$$\n\boxed{16}\n$$", "Explore further into the properties of $\phi(n)$, and unlock the hidden order within integers—one of mathematics’ most elegant frontiers."]