Solution: Factor 120 into primes: $120 = 2^3 \times 3 \times 5$. The distinct primes are 2, 3, and 5. Their sum is $2 + 3 + 5 = 10$. \boxed{10}

["Mastering Prime Factorization: The Simple Solution to Factor 120 and Find the Sum of Its Prime Facts", "Understanding how to factor numbers into primes is a fundamental skill in mathematics that unlocks deeper insights into number theory, algebra, and problem-solving. A key example is factoring 120 into prime components — a process that reveals its distinct prime factors and helps calculate meaningful values like their sum.", "### What Does Prime Factorization Mean?", "Prime factorization is the process of expressing a composite number as the product of prime numbers. Every integer greater than 1 is either a prime itself or can be broken down into a unique product of primes, thanks to the Fundamental Theorem of Arithmetic.", "### The Prime Factorization of 120", "Let’s break down 120 step by step:", "1. Start dividing by the smallest prime, 2:\n (120 ÷ 2 = 60), (60 ÷ 2 = 30), (30 ÷ 2 = 15)\n This gives (2^3) (since 2 divides exactly 3 times).", "2. Next, divide by the next smallest prime, 3:\n (15 ÷ 3 = 5)\n Now we have (3^1).", "3. Finally, we’re left with 5, which is itself a prime number.", "Putting it all together:\n[\n120 = 2^3 \ imes 3 \ imes 5\n]", "The distinct prime factors of 120 are the unique primes involved: 2, 3, and 5.", "### The Sum of Distinct Prime Factors", "To find a meaningful number derived from these primes, simply sum them:\n[\n2 + 3 + 5 = 10\n]", "This sum, often called the sum of distinct prime factors, is useful in number theory, cryptography, and algorithmic problems — especially when analyzing factors for optimization or pattern recognition.", "### Why This Matters", "Knowing how to factor numbers and extract their prime building blocks empowers you to:\n- Simplify fractions more efficiently\n- Solve equations involving divisors\n- Understand the structure of integers deeper\n- Apply knowledge in coding or data analysis", "### Conclusion", "Factoring 120 as (2^3 \ imes 3 \ imes 5) gives us the distinct primes 2, 3, and 5 — whose sum is 10. This foundational concept in math opens doors to more complex mathematical reasoning and practical applications.", "[\n\boxed{10}\n]", "---", "Keywords: prime factorization of 120, distinct prime factors, sum of prime numbers, factorization definition, mathematical fundamentals, prime decomposition"]