Thus, the sum of the distinct prime factors is $\boxed{16}$.
["Title: Exploring the Sum of Distinct Prime Factors: When It Equals 16", "Understanding number theory offers fascinating insights into the building blocks of arithmetic—especially prime factors. One intriguing mathematical puzzle arises when we examine the sum of distinct prime factors of a number that equals 16. This article explores the concept, identifies the number(s) involved, and explains why the sum of their distinct prime factors equals 16.", "### What Are Prime Factors?", "A prime factor of a number is a prime number that divides evenly into that number. Every integer greater than 1 can be uniquely expressed as a product of primes—this is known as the Fundamental Theorem of Arithmetic. Among these prime factors, only the distinct ones matter when calculating their sum.", "For example, consider the number 100. Its prime factorization is:", "[\n100 = 2^2 \ imes 5^2\n]", "The distinct prime factors here are 2 and 5, and their sum is:", "[\n2 + 5 = 7\n]", "But what if the sum equals 16? Let’s explore that.", "---", "### Finding Numbers Where the Sum of Distinct Prime Factors Is 16", "We seek a number whose prime factors, when added together without repetition, sum to 16.", "Let’s analyze combinations of distinct primes that add up to 16:", "1. 2 + 3 + 5 + 7 = 17 → Too high\n2. 2 + 3 + 11 = 16 → Valid!\n3. 2 + 5 + 11 = 18 → Too high\n4. 3 + 13 = 16 → Valid!\n5. 3 + 5 + 7 = 15 → Close but not 16\n6. 5 + 11 = 16 → Valid!\n7. 13 + 3 = 16 → Same as earlier, just reordered", "So the valid sets of distinct prime numbers whose sum is 16 are:", "- {2, 3, 11}\n- {3, 13}\n- {5, 11}", "Let’s examine one example in depth: take 3 + 13 = 16.", "The number must include only these two primes as distinct factors—no more, no less. So consider:", "[\nn = 3 \ imes 13 = 39\n]", "Prime factorization:\n[\n39 = 3 \ imes 13 \quad (\ ext{Both primes, distinct})\n]", "Sum of distinct prime factors:\n[\n3 + 13 = \boxed{16}\n]", "---", "### Why This Matters in Number Theory and Number Puzzles", "While the sum itself may seem abstract, understanding such properties enriches the study of integers and their structure. These types of problems appear in:", "- Competitive mathematics: Challenging puzzles in Olympiad-level problems\n- Computational number theory: Algorithms that factorize numbers efficiently\n- Educational tools: Teaching students about prime decomposition and number theory fundamentals", "---", "### Numbers with Sum of Distinct Prime Factors Equal to 16 Are Special", "Although several combinations yield 16, each corresponds to different products. The key constraint is that only the distinct primes contribute to the sum, ignoring exponents. So:", "- ( 3 \ imes 13 = 39 )\n- ( 39 ) restricts factors only to 3 and 13\n- ( 3 \ imes 13^2 = 507 ), still sum = 16\n- ( 5 \ imes 11 = 55 ), also yields sum 16", "Each gives a unique number, but all share the elegant property: sum of distinct primes = 16.", "---", "### Conclusion: A Harmonious Sum in Mathematics", "The statement “the sum of the distinct prime factors is $\boxed{16}$” illuminates the beauty of prime decomposition. Whether through 39, 55, or 507, these numbers exemplify how small sets of primes can define identities and relationships central to number theory. Exploring such sums deepens appreciation for the hidden order within the integers.", "---", "### Further Reading", "- Prime Factorization Techniques\n- Fundamental Theorem of Arithmetic\n- Applications of Number Theory in Cryptography\n- Integer Decomposition Problems", "Start your journey into prime numbers today—and discover how simply adding distinct factors can reveal profound mathematical harmony.", "---", "Keywords: sum of distinct prime factors, prime factor sum equals 16, number theory, prime factorization, distinct primes, mathematical puzzles, Fundamental Theorem of Arithmetic, 39 sum prime factors, 55, 507 number examples."]
