Question: What is the sum of all of the positive odd divisors of $ 180 $?
["# What is the Sum of All Positive Odd Divisors of 180?", "Understanding the sum of positive odd divisors of a number is essential in number theory and has practical applications in mathematics and cryptography. A commonly asked question is: What is the sum of all positive odd divisors of 180? In this article, we explain step-by-step how to compute this sum, highlighting key mathematical concepts like prime factorization, divisor functions, and selective filtering of divisors.", "## Step 1: Prime Factorization of 180", "To find the odd divisors of 180, start by determining its prime factorization:", "[\n180 = 2^2 \ imes 3^2 \ imes 5^1\n]", "Since only odd divisors are required, we ignore the factor of $2^2$. The odd part of 180 is:", "[\n3^2 \ imes 5^1\n]", "## Step 2: Formula for Sum of Divisors", "For a number $n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$, the sum of all positive divisors is given by:", "[\n\sigma(n) = (1 + p_1 + p_1^2 + \cdots + p_1^{e_1})(1 + p_2 + \cdots + p_2^{e_2}) \cdots (1 + p_k + \cdots + p_k^{e_k})\n]", "This multiplicative formula lets us compute the sum efficiently using each prime’s contributions.", "For the odd part $3^2 \ imes 5^1$, the sum of its divisors is:", "[\n\sigma_{\ ext{odd}}(180) = (1 + 3 + 3^2)(1 + 5)\n]", "## Step 3: Compute Each Factor", "Calculate each factor separately:", "- $1 + 3 + 9 = 13$\n- $1 + 5 = 6$", "Multiply them together:", "[\n13 \ imes 6 = 78\n]", "## Step 4: List the Odd Divisors (Optional Verification)", "To confirm, list all positive odd divisors of 180 by only using powers of $3$ and $5$:", "Possible exponents:\n- $3^0, 3^1, 3^2$\n- $5^0, 5^1$", "All combinations yield these odd divisors:", "[\n1,\ 3,\ 9,\ 5,\ 15,\ 45\n]", "Sum them:", "[\n1 + 3 + 9 + 5 + 15 + 45 = 78\n]", "## Conclusion", "The sum of all positive odd divisors of $180$ is $78$. By eliminating even factors through exclusion of $2^k$, and applying the divisor sum formula to the odd part $3^2 \ imes 5$, we efficiently determine this result. This method highlights the power of prime factorization and selective divisor processing in number theory.", "## Key Takeaways", "- Focus only on odd prime factors to find odd divisors.\n- Use the divisor sum formula to avoid listing all factors.\n- Verify by direct computation or listing.", "Whether you're solving math problems or exploring number theory, understanding how to compute the sum of odd divisors is both useful and elegant!", "---", "Keywords: sum of odd divisors of 180, divisor function, prime factorization, number theory, calculate odd divisors, 180 divisors, math tutorial, divisor sum formula", "Meta Description: Find the sum of all positive odd divisors of 180 using prime factorization and the divisor sum formula. Learn step-by-step with verified results and missing factor exclusion methods."]
