What is the sum of all the even divisors of 60?

["# The Sum of All Even Divisors of 60: A Step-by-Step Breakdown", "When exploring number theory, an interesting question often arises: What is the sum of all the even divisors of 60? Understanding this concept not only helps strengthen foundational math skills but also offers insight into divisor behavior and prime factorization. In this article, we’ll explore the proper mathematical approach to find the sum of all even divisors of 60 — and provide a clear, step-by-step explanation so you can confidently solve similar problems.", "---", "## What Are Divisors?", "A divisor of a number is any integer that divides that number without leaving a remainder. For example, the divisors of 60 include:\n1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.", "Divisors can be classified as even or odd, based on their divisibility by 2.", "---", "## Why Focus on Even Divisors?", "Even divisors are those divisible by 2. By isolating even divisors, we simplify problems involving factors, sums, and multiplications—especially useful in number theory, combinatorics, and algorithm design.", "---", "## Quick Insight: All Divisors of 60", "To find the even divisors, first list all divisors of 60:", "All divisors of 60:\n1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60", "Now, identify which are even:\n2, 4, 6, 10, 12, 20, 30, 60", "These are the even divisors of 60. But calculating their sum manually is tedious — and error-prone. Instead, we’ll use a smarter method.", "---", "## How to Find the Sum of Even Divisors Efficiently", "### Step 1: Prime Factorization of 60", "First, factor 60 into its prime components:", "[\n60 = 2^2 \ imes 3^1 \ imes 5^1\n]", "---", "### Step 2: Sum of All Divisors Formula", "The sum of all positive divisors of a number ( n = p^a \ imes q^b \ imes r^c ) is given by:", "[\n\sigma(n) = (1 + p + p^2 + \dots + p^a)(1 + q + q^2 + \dots + q^b)(1 + r + r^2 + \dots + r^c)\n]", "For 60:\n[\n\sigma(60) = (1 + 2 + 2^2)(1 + 3)(1 + 5) = (1 + 2 + 4)(4)(6) = 7 \ imes 4 \ imes 6 = 168\n]", "So, the sum of all divisors of 60 is 168.", "---", "### Step 3: Separate Even and Odd Divisors", "An essential insight:\nEven divisors must include at least one factor of 2. So, every even divisor of 60 can be written as:\n[\n2^a \ imes 3^b \ imes 5^c \quad \ ext{where } a \geq 1,\ 0 \leq b \leq 1,\ 0 \leq c \leq 1\n]", "But since ( 2^2 ) is the highest power of 2 in 60, ( a = 1 ) or ( a = 2 ).", "To count and sum only even divisors, we exclude divisors where the exponent of 2 is zero.", "Thus, split the sum into:", "[\n\ ext{Sum of even divisors} = \left( \sum_{\ ext{even } a} 2^a \right) \ imes (1 + 3) \ imes (1 + 5)\n]", "Where ( a = 1 ) or ( 2 )", "Compute each part:", "- Odd part from 3: ( 1 + 3 = 4 )\n- Odd part from 5: ( 1 + 5 = 6 )\n- Even powers of 2: ( 2^1 + 2^2 = 2 + 4 = 6 )", "Now multiply:", "[\n6 \ imes 4 \ imes 6 = 144\n]", "---", "## Final Answer: Sum of All Even Divisors of 60", "[\n\boxed{144}\n]", "---", "## Verification: Listed and Summed", "List all even divisors:\n2, 4, 6, 10, 12, 20, 30, 60", "Add them:\n2 + 4 = 6\n6 + 6 = 12\n12 + 10 = 22\n22 + 12 = 34\n34 + 20 = 54\n54 + 30 = 84\n84 + 60 = 144", "✅ Matches our calculation.", "---", "## Why This Method Works", "By factoring out the 2 and restricting its exponent to 1 or 2, we systematically isolate even divisors while efficiently computing their total without listing every one. This approach scales well to much larger numbers.", "---", "## Key Takeaways", "- Even divisors are multiples of 2.\n- Prime factorization enables quick, accurate sum calculations.\n- Using the divisor sum formula and conditional exponent ranges streamlines computations.\n- Always verify via direct addition when possible.", "Understanding even divisors enhances number sense and prepares you for advanced math topics like multiplicative functions and modular arithmetic.", "---", "## Summary", "The sum of all even divisors of 60 is 144 — derived elegantly using prime factorization and divisor sum principles. Whether for homework, exams, or personal learning, mastering this method unlocks deeper insight into number theory.", "---", "Keywords: sum of even divisors, even divisors of 60, divisor sum formula, prime factorization, number theory, math tutorial\nAlso search: sum of even divisors of 60 explanation, how to find divisor sums, 60 divisors list"]