Thus, the sum of all positive divisors of $ 1440 $ congruent to $ 1 \pmod{4} $ is $ \boxed{60} $.
["Sum of All Positive Divisors of 1440 Congruent to (1 \pmod{4}): A Detailed Calculation", "The number 1440 is a rich source of mathematical interest, particularly when analyzing its positive divisors with specific modular properties. This article explores the sum of all positive divisors of 1440 that are congruent to (1) modulo (4), revealing a precise value: ( \boxed{60} ).", "### Understanding the Problem", "We are tasked with computing:", "[\nS = \sum_{\substack{d \mid 1440 \ d \equiv 1 \pmod{4}}} d\n]", "That is, sum all proper divisors of 1440 that leave a remainder of 1 when divided by 4.", "### Step 1: Prime Factorization of 1440", "Begin by factoring 1440 into its prime components:", "[\n1440 = 144 \ imes 10 = (12^2) \ imes 10 = (2^2 \cdot 3)^2 \cdot (2 \cdot 5) = 2^5 \cdot 3^2 \cdot 5^1\n]", "So, the prime factorization is:", "[\n1440 = 2^5 \cdot 3^2 \cdot 5\n]", "### Step 2: Generate All Divisors", "Divisors of 1440 are of the form:", "[\n2^a \cdot 3^b \cdot 5^c \quad \ ext{where } 0 \leq a \leq 5,; 0 \leq b \leq 2,; 0 \leq c \leq 1\n]", "This gives (6 \ imes 3 \ imes 2 = 36) total positive divisors. We do not list all, but instead identify those satisfying (d \equiv 1 \pmod{4}).", "### Step 3: Characterize Divisors Modulo 4", "Since we care about (d \equiv 1 \pmod{4}), we analyze the divisibility by powers of 2:", "- If (a \geq 2), then (d) is divisible by (4), so (d \equiv 0 \pmod{4}) — not acceptable.\n- If (a = 1), (d = 2 \cdot (3^b \cdot 5^c)). Then modulo 4, since (3^b \cdot 5^c) is odd, doubling it gives (2 \pmod{4}) — so (d \equiv 2 \pmod{4}), not (1).\n- If (a = 0), (d = 3^b \cdot 5^c), and now (d) is odd — we analyze these further.", "So only divisors with (a = 0) (i.e., odd divisors of 1440) can be ( \equiv 1 \pmod{4} ). These are:", "[\nd = 3^b \cdot 5^c \quad \ ext{with } b = 0,1,2,; c = 0,1\n]", "List them:", "- (3^0 \cdot 5^0 = 1) → (1 \equiv 1 \pmod{4})\n- (3^1 \cdot 5^0 = 3) → (3 \equiv 3 \pmod{4})\n- (3^2 \cdot 5^0 = 9) → (9 \equiv 1 \pmod{4})\n- (3^0 \cdot 5^1 = 5) → (5 \equiv 1 \pmod{4})\n- (3^1 \cdot 5^1 = 15) → (15 \equiv 3 \pmod{4})\n- (3^2 \cdot 5^1 = 45) → (45 \equiv 1 \pmod{4})", "Thus, the divisors of 1440 congruent to (1 \pmod{4}) are: (1, 9, 5, 45).", "### Step 4: Compute the Sum", "Now sum these valid divisors:", "[\n1 + 9 + 5 + 45 = 60\n]", "### Step 5: Why This Sum Matters", "This result highlights the utility of modular arithmetic in number theory, particularly in studying divisor functions under congruence constraints. The restriction to (a=0) is crucial — only odd divisors survive modulo 4, reducing the search space significantly.", "### Final Answer", "[\n\boxed{60}\n]", "---", "Conclusion", "The sum of all positive divisors of (1440) that are congruent to (1 \pmod{4}) is exactly (60), confirmed through systematic factorization and modular analysis. This approach exemplifies elegant number-theoretic computation."]
