Thus, the sum of the distinct prime factors is \(\boxed{12}\).
["### Thus, the Sum of the Distinct Prime Factors Is Boxed: (\boxed{12})", "When analyzing a composite number, identifying its prime factors plays a crucial role in number theory and practical applications like cryptography, simplification of fractions, and understanding factorization patterns. In this article, we explore a clear example where the sum of a number’s distinct prime factors yields a simple yet insightful result—the sum equals (\boxed{12}).", "#### What Are Distinct Prime Factors?", "Distinct prime factors of a number are the unique prime integers that multiply together to produce the original number. Unlike repeated factors, distinct primes ensure each contributes once to the factorization. For instance, if (N = 180), its prime factorization is (180 = 2^2 \ imes 3^2 \ imes 5), so the distinct prime factors are 2, 3, and 5.", "#### The Example: Why Sum Equals 12?", "Consider a specific composite number where the sum of its distinct prime factors is exactly (\boxed{12}). Take (N = 105). Its full prime factorization is:", "[\n105 = 3 \ imes 5 \ imes 7\n]", "Here, the distinct prime factors are 3, 5, and 7. Adding them together:", "[\n3 + 5 + 7 = 15\n]", "Wait—this does not yield 12. So which number gives a sum of 12? Let’s reverse-engineer: suppose a number has distinct primes whose sum is 12. Try combinations of small primes adding to 12:", "- (2 + 3 + 7 = 12)\n- (2 + 5 + 5) is invalid (duplicate primes)\n- (3 + 4)—4 isn’t prime\n- (5 + 7 = 12), but that’s two primes", "Only (2 + 3 + 7 = 12) uses three distinct primes. So consider (N = 2 \ imes 3 \ imes 7 = 42). Let’s verify:", "- Distinct prime factors: 2, 3, 7\n- Their sum: (2 + 3 + 7 = 12)", "Thus, the sum of the distinct prime factors of 42 is indeed:", "[\n\boxed{12}\n]", "#### Why This Matters?", "Understanding such examples deepens insight into how prime decomposition encodes number properties. The sum of distinct primes serves as a foundational concept in:", "- Cryptography: Prime sieves and factor checks rely on understanding factor sets.\n- Sustainable factorization: Efficient algorithms use prime uniqueness to reduce computational complexity.\n- Mathematical education: Clear examples like sum = 12 simplify abstract theories for learners.", "#### Conclusion", "The statement (\boxed{12})—the sum of distinct prime factors—holds meaningful value in both computation and concept. It highlights how small primes can combine to form insightful arithmetic identities.", "Whether simplifying fractions, testing primality, or teaching number theory, recognizing when such sums equal 12 supports clear and correct reasoning. This number isn’t just a boxed answer—it embodies a key principle in prime analysis.", "---", "Keywords: distinct prime factors, sum of prime factors, number theory example, boxed result, mathematical identity, ( \boxed{12} ) explanation", "---", "By examining precise examples, this article reinforces the importance of distinct prime decomposition and why (\boxed{12}) distinctly represents the sum of primes for a well-understood number."]
