Question: What is the sum of all the squares of the divisors of $ 12 $?
["What is the Sum of All the Squares of the Divisors of 12?", "When exploring number theory, one fascinating question is: What is the sum of all the squares of the divisors of 12? This problem combines divisors, squaring, and summation, making it a great example of arithmetic properties and divisor functions.", "### Understanding the Problem", "To find the sum of the squares of all positive divisors of 12, we must first identify all divisors of 12, then square each, and finally compute their total.", "### Step 1: Find the Divisors of 12", "The positive divisors of 12 are the integers that divide 12 without leaving a remainder. We list them:", "[\n1,\ 2,\ 3,\ 4,\ 6,\ 12\n]", "Verification:\n12 ÷ 1 = 12\n12 ÷ 2 = 6\n12 ÷ 3 = 4\n12 ÷ 4 = 3\n12 ÷ 6 = 2\n12 ÷ 12 = 1\nIndeed, these are all divisors.", "### Step 2: Square Each Divisor", "Now calculate the square of each divisor:", "[\n1^2 = 1 \\n2^2 = 4 \\n3^2 = 9 \\n4^2 = 16 \\n6^2 = 36 \\n12^2 = 144\n]", "### Step 3: Sum the Squares", "Add them together:", "[\n1 + 4 + 9 + 16 + 36 + 144 = 210\n]", "### Final Result", "Thus, the sum of the squares of all divisors of 12 is:", "[\n\boxed{210}\n]", "---", "### Why This Matter: The Algebraic Insight", "Beyond computation, this sum connects deeply to number theory. For any positive integer $ n $ with prime factorization $ n = p_1^{e_1}p_2^{e_2} \cdots p_k^{e_k} $, the sum of the squares of its divisors is given by:", "[\n\prod_{i=1}^k \left(1 + p_i^2 + p_i^4 + \cdots + p_i^{2e_i}\right) = \prod_{i=1}^k \frac{p_i^{2(e_i+1)} - 1}{p_i^2 - 1}\n]", "For $ n = 12 = 2^2 \cdot 3^1 $:", "[\n\ ext{Sum of squares} = (1 + 2^2 + 2^4)(1 + 3^2) = (1 + 4 + 16)(1 + 9) = 21 \cdot 10 = 210\n]", "This confirms our earlier result elegantly.", "---", "### Conclusion", "The sum of the squares of the divisors of 12 is a simple yet powerful example illustrating how divisor functions and prime factorization power calculations unify in integer theory. Whether for math enthusiasts, students, or educators, such problems deepen understanding of number properties and celebrate the beauty of mathematics.", "Keywords: sum of squares of divisors, divisors of 12, number theory, divisor function, mathematical computation, 12 factors, divisor sum formula."]
