\phi(48) = 48 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right) = 48 \cdot \frac{1}{2} \cdot \frac{2}{3} = 48 \cdot \frac{1}{3} = 16

["Understanding φ(48): Unlocking Euler’s Totient Function with a Simple Product Formula", "When exploring number theory, one of the fascinating elements is Euler’s totient function, denoted φ(n). This function counts the number of positive integers less than or equal to n that are coprime to n — in simpler terms, it measures how many integers less than 48 share no common factors with 48 other than 1.", "In this article, we’ll break down an elegant calculation:\n[\n\phi(48) = 48 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right) = 48 \cdot \frac{1}{2} \cdot \frac{2}{3} = 48 \cdot \frac{1}{3} = 16\n]", "### What Is Euler’s Totient Function?", "Euler’s totient function φ(n) is defined as:\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_k}\right)\n]\nwhere ( p_1, p_2, \dots, p_k ) are the distinct prime factors of ( n ).", "This formula comes from the principle that we subtract fractions of numbers divisible by each prime factor from the total count up to n.", "### Prime Factorization of 48", "To apply the formula correctly, we first factor 48 into its prime components:\n[\n48 = 2^4 \ imes 3\n]\nThe distinct prime factors are 2 and 3.", "### Applying the Totient Formula", "Substituting into Euler’s formula:\n[\n\phi(48) = 48 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\n]", "Let’s compute each term step-by-step:\n- ( 1 - \frac{1}{2} = \frac{1}{2} )\n- ( 1 - \frac{1}{3} = \frac{2}{3} )", "Now multiply:\n[\n\phi(48) = 48 \ imes \frac{1}{2} \ imes \frac{2}{3}\n]\n[\n= 48 \ imes \frac{1 \ imes 2}{2 \ imes 3} = 48 \ imes \frac{2}{6} = 48 \ imes \frac{1}{3} = 16\n]", "Thus,\n[\n\phi(48) = 16\n]", "### Why Is This Result Meaningful?", "This means there are 16 integers between 1 and 48 that are relatively prime to 48 — that is, their greatest common divisor (GCD) with 48 is 1. These numbers are:\n1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47", "Such values are critical in cryptography, modular arithmetic, and number theory applications, particularly in RSA encryption where totient calculations ensure secure key generation.", "### Summary", "By factoring 48 into primes and applying Euler’s totient formula, we efficiently calculate:\n[\n\phi(48) = 16\n]\nThis simple but powerful computation illustrates the elegance of number theory and forms a foundational concept for advanced mathematical study and practical technology.", "If you're interested in exploring more about φ(n) and its applications, this function continues to reveal rich patterns in the distribution of coprime numbers.", "---", "Keywords: Euler’s totient function φ(48), calculation of φ(48), prime factorization of 48, coprime numbers, number theory, cryptography applications, φ(48) = 16.\nMeta Description: Learn how to compute Euler’s totient function φ(48) step-by-step, using prime factorization and the multiplicative formula, resulting in φ(48) = 16. Ideal for students and hobbyists in number theory."]