We seek the sum of all positive divisors $ d \mid 1440 $ such that $ d \equiv 1 \pmod{4} $.
["# We Seek the Sum of All Positive Divisors $ d \mid 1440 $ Such That $ d \equiv 1 \pmod{4} $", "Understanding the divisors of a number is fundamental in number theory, revealing deep structural properties. This article explores a specific problem: finding the sum of all positive divisors $ d $ of $ 1440 $ that satisfy the condition $ d \equiv 1 \pmod{4} $. This type of inquiry is valuable both mathematically and algorithmically, offering insight into modular arithmetic and factorization.", "---", "## Understanding the Problem", "We want to compute:", "$$\nS = \sum_{\substack{d \mid 1440 \ d \equiv 1 \pmod{4}}} d\n$$", "That is, sum all positive divisors $ d $ of $ 1440 $ such that when $ d $ is divided by $ 4 $, the remainder is $ 1 $.", "Divisors of $ 1440 $ are all integers $ d $ satisfying $ 1 \leq d \leq 1440 $, $ d \mid 1440 $. But instead of listing all $ 48 $ divisors, we focus only on those congruent to $ 1 \mod 4 $.", "---", "## Step 1: Prime Factorization of 1440", "Start by factoring $ 1440 $:", "$$\n1440 = 144 \ imes 10 = (12^2) \ imes (2 \ imes 5) = (2^4 \cdot 3^2) \cdot (2 \cdot 3 \cdot 5) = 2^5 \cdot 3^2 \cdot 5\n$$", "So,\n$$\n1440 = 2^5 \cdot 3^2 \cdot 5\n$$", "The total number of positive divisors is $ (5+1)(2+1)(1+1) = 6 \cdot 3 \cdot 2 = 36 $. There are 36 divisors to consider, but we only want those $ d $ with $ d \equiv 1 \pmod{4} $.", "---", "## Step 2: Characterizing Divisors Modulo 4", "We seek divisors $ d = 2^a \cdot 3^b \cdot 5^c $, where:\n- $ 0 \leq a \leq 5 $\n- $ 0 \leq b \leq 2 $\n- $ 0 \leq c \leq 1 $", "and $ d \equiv 1 \pmod{4} $.", "Note:\n- If $ a \geq 2 $, then $ d $ is divisible by $ 4 $, so $ d \equiv 0, 2 \pmod{4} $, hence not $ \equiv 1 $.\n- If $ a = 1 $, $ d $ is even but not divisible by 4, so $ d \equiv 2 \pmod{4} $ — still not $ \equiv 1 $.\n- Only when $ a = 0 $ (i.e., $ d $ odd) can $ d \equiv 1 \pmod{4} $, since powers of 2 contribute only $ 1 \mod 4 $ or $ 2 \mod 4 $ otherwise.", "Thus, $ d \equiv 1 \pmod{4} $ implies $ d $ is odd, so $ a = 0 $.", "So we restrict to odd divisors of $ 1440 $, i.e., those with $ a = 0 $. From the factorization, this means $ 3^b \cdot 5^c $, where $ b = 0,1,2 $, $ c = 0,1 $. Total odd divisors: $ 3 \cdot 2 = 6 $.", "List them:\n- $ 3^0 \cdot 5^0 = 1 $\n- $ 3^1 \cdot 5^0 = 3 $\n- $ 3^2 \cdot 5^0 = 9 $\n- $ 3^0 \cdot 5^1 = 5 $\n- $ 3^1 \cdot 5^1 = 15 $\n- $ 3^2 \cdot 5^1 = 45 $", "So odd divisors: $ 1, 3, 5, 9, 15, 45 $", "Now compute each modulo 4:\n- $ 1 \mod 4 = 1 $\n- $ 3 \mod 4 = 3 $\n- $ 5 \mod 4 = 1 $\n- $ 9 \mod 4 = 1 $\n- $ 15 \mod 4 = 3 $\n- $ 45 \mod 4 = 1 $", "So $ d \equiv 1 \pmod{4} $ among odd divisors: $ 1, 5, 9, 45 $", "---", "## Step 3: Sum the Valid Divisors", "$$\nS = 1 + 5 + 9 + 45 = 60\n$$", "---", "## Verification via Multiplicative Structure", "The multiplicative nature of divisors helps confirm: the set of divisors $ d \mid 1440 $, $ d \equiv 1 \pmod{4} $, corresponds exactly to $ 3^b \cdot 5^c $ with $ b \in {0,1,2}, c \in {0,1} $, and all such are $ \equiv 1 \pmod{4} $ because they are odd and $ <br/>\not\equiv 3 \mod 4 $. Since $ 3 \equiv 3 $, $ 3^2 = 9 \equiv 1 \mod 4 $, and $ 5 \equiv 1 \mod 4 $, the combinations multiply to values $ \equiv 1 \mod 4 $:", "- $ 3^0 \cdot 5^0 = 1 \equiv 1 $\n- $ 3^1 \cdot 5^0 = 3 \equiv 3 $\n- $ 3^2 \cdot 5^0 = 9 \equiv 1 $\n- $ 3^0 \cdot 5^1 = 5 \equiv 1 $\n- $ 3^1 \cdot 5^1 = 15 \equiv 3 $\n- $ 3^2 \cdot 5^1 = 45 \equiv 1 $", "So again, only $ 1, 5, 9, 45 $ qualify.", "Sum: $ 1 + 5 + 9 + 45 = 60 $", "---", "## Conclusion", "The sum of all positive divisors $ d $ of $ 1440 $ satisfying $ d \equiv 1 \pmod{4} $ is $ \boxed{60} $. This result combines efficient modular filtering with structural analysis of multiplicative divisor classes, typical in number theory and computational math.", "For algorithmically inclined readers or programmers, this approach scales: systematically eliminating divisors by exponent constraints and using modular arithmetic ensures correctness and efficiency. Such methods extend to divisor sums with other congruence conditions, forming a cornerstone of analytic number theory.", "---", "Keywords: divisors of 1440, modulo 4, sum of divisors, $ d \equiv 1 \pmod{4} $, number theory, multiplicative functions, factorization, $ 1440 = 2^5 \cdot 3^2 \cdot 5 $, sum $ d \equiv 1 \pmod{4} $, mathematical decomposition, modular arithmetic."]
