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Home » Shocked After Using XXXm—Here’s the Surprising Result You Won’t Ignore! - Dyverse

Shocked After Using XXXm—Here’s the Surprising Result You Won’t Ignore! - Dyverse

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Final answer: $ oxed{rac{168}{13}} $Question: What is the remainder when the sum of the first 12 even numbers after 2020 is divided by 11?

📌 Solution: The first 12 even numbers after 2020 are $2022, 2024, \ldots, 2044$. The sum of an arithmetic sequence is $S = rac{n}{2}(a_1 + a_n)$. Here, $n = 12$, $a_1 = 2022$, $a_n = 2044$. Thus, $S = rac{12}{2}(2022 + 2044) = 6 imes 4066 = 24396$. Dividing 24396 by 11: $11 imes 2216 = 24376$, so the remainder is $24396 - 24376 = 20$. However, $20 \mod 11 = 9$. Therefore, the remainder is $oxed{9}$.

📌 Question: What is the least common multiple of 2048 and 1024?

📌 Solution: The prime factorization of 2048 is $2^{11}$, and 1024 is $2^{10}$. The LCM of two numbers is the product of the highest powers of all primes present. Thus, $ ext{LCM}(2048, 1024) = 2^{11} = 2048$. The answer is $oxed{2048}$.

📌 Question: What is the base-ten number represented by the base-eight number 315?

📌 Solution: Converting $315_8$ to decimal: $3 imes 8^2 + 1 imes 8^1 + 5 imes 8^0 = 3 imes 64 + 1 imes 8 + 5 imes 1 = 192 + 8 + 5 = 205$. The base-ten equivalent is $oxed{205}$.

📌 Question: How many of the 100 smallest positive integers are congruent to 3 mod 7?

📌 Solution: Numbers congruent to 3 mod 7 are $3, 10, 17, \ldots, 3 + 7k$. The largest such number $\leq 100$ satisfies $3 + 7k \leq 100 \Rightarrow k \leq rac{97}{7} pprox 13.857$. Thus, $k = 0, 1, \ldots, 13$, giving 14 numbers. The count is $oxed{14}$.

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