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Home » \( f(1) = a(1) + b = a + b \) - AIKO, infinite ways to autonomy.

\( f(1) = a(1) + b = a + b \) - AIKO, infinite ways to autonomy.

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 This is an arithmetic series: sum = \(rac{n}{2} imes ( ext{first} + ext{last}) = rac{20}{2} imes (2 + 40) = 10 Ã 42 = 420\).

📌 #### 420Question: How many integers between 1 and 500 are congruent to 2 mod 7?

📌 Solution: To find the count of integers $ n $ such that $ 1 \leq n \leq 500 $ and $ n \equiv 2 \pmod{7} $, we note the sequence starts at 2 and increases by 7 each time: $ 2, 9, 16, \dots $. The general term is $ 7k + 2 $. Solving $ 7k + 2 \leq 500 $ gives $ k \leq rac{498}{7} pprox 71.14 $, so $ k = 0, 1, \dots, 71 $. This yields $ 72 $ integers. oxed{72}

📌 Question: What is the greatest common divisor of 210 and 294, representing shared efficiency metrics in clean energy systems?

📌 Solution: Factorize $ 210 = 2 \cdot 3 \cdot 5 \cdot 7 $ and $ 294 = 2 \cdot 3 \cdot 7^2 $. The GCD is the product of the smallest powers of common primes: $ 2^1 \cdot 3^1 \cdot 7^1 = 42 $. oxed{42}

📌 Question: What is the sum of all positive divisors of 120 that are multiples of 5, reflecting resource allocation in smart city initiatives?

📌 Solution: The divisors of 120 are $ 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 $. Those divisible by 5 are $ 5, 10, 15, 20, 30, 40, 60, 120 $. Summing them: $ 5 + 10 + 15 + 20 + 30 + 40 + 60 + 120 = 290 $. oxed{290}Question: Let $ x $ and $ y $ be positive real numbers representing the bandwidths (in Gbps) of two parallel data streams. Find the minimum value of

📌 rac{x^2 + y^2}{xy} + rac{xy}{x^2 + y^2}

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Thus, the length of the diagonal is:

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