Question: What is the sum of all positive divisors of \( 360 \) that are perfect squares?
["# What Is the Sum of All Positive Divisors of 360 That Are Perfect Squares?", "Understanding the sum of positive divisors of a number can reveal valuable insights into its number theory properties. One particularly interesting case involves finding the sum of all positive divisors of 360 that are perfect squares. Whether you're a student exploring divisors, a math enthusiast, or someone studying number theory fundamentals, this guide will walk you through the process step-by-step.", "## Step 1: Factorize 360 into Its Prime Components", "To find the divisors of 360, begin with its prime factorization:", "[\n360 = 2^3 \ imes 3^2 \ imes 5^1\n]", "This decomposition is essential because it lets us systematically generate all divisors by choosing powers of 2, 3, and 5 within their respective ranges.", "## Step 2: Identify All Divisors That Are Perfect Squares", "A perfect square divisor must have even exponents in its prime factorization. Why? Because a square number has prime factors raised to even powers (e.g., ( 2^2, 3^4 ) — but not ( 2^3, 3^1 )).", "Given ( 360 = 2^3 \ imes 3^2 \ imes 5^1 ), for a divisor to be a perfect square, the exponents of each prime must satisfy:", "- Exponent of 2: even values ≤ 3 → possible values: 0 or 2\n- Exponent of 3: even values ≤ 2 → possible values: 0 or 2\n- Exponent of 5: even values ≤ 1 → only 0 (since 1 is odd)", "Thus, only combinations with exponents (0 or 2) for 2, (0 or 2) for 3, and 0 for 5 are valid.", "### List of Perfect Square Divisors of 360", "From the allowed exponents:", "- ( 2^0 \ imes 3^0 \ imes 5^0 = 1 )\n- ( 2^2 \ imes 3^0 \ imes 5^0 = 4 )\n- ( 2^0 \ imes 3^2 \ imes 5^0 = 9 )\n- ( 2^2 \ imes 3^2 \ imes 5^0 = 36 )", "These are the only divisors of 360 that are perfect squares.", "## Step 3: Calculate the Sum of These Divisors", "Now sum them:", "[\n1 + 4 + 9 + 36 = 50\n]", "## Conclusion: Final Answer", "The sum of all positive divisors of ( 360 ) that are perfect squares is 50.", "This result not only showcases the elegance of divisor function analysis but also demonstrates how prime factorization and parity of exponents help isolate special subsets—like perfect squares—within the full set of divisors.", "---", "### Bonus: Practice and Further Reading", "- Explore how prime factorization underpins the divisor sum formula.\n- Try the same approach for other numbers like 144 or 1024.\n- Learn about sigma functions ( \sigma_k(n) ), which generalize sums of divisors with exponents.", "Understanding such properties lays a strong foundation for number theory, cryptography, and algorithm design.", "---", "Keywords: sum of positive divisors of 360, perfect square divisors of 360, divisors of 360, number theory, prime factorization, divisor sum formula, σ-function."]
