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Home » Therefore, the least common multiple is \(\boxed24\). - Leaselab

Therefore, the least common multiple is \(\boxed24\). - Leaselab

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 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 $oxed{14}$.

📌 Question: What is the greatest common divisor of 2025 and 1024?

📌 Solution: The prime factorization of 2025 is $3^4 imes 5^2$, and 1024 is $2^{10}$. Since there are no common prime factors, the GCD is 1. The answer is $oxed{1}$.

📌 Question: What is the remainder when $1^2 + 2^2 + \cdots + 15^2$ is divided by 13?

📌 Solution: The sum of squares formula gives $S = rac{15 imes 16 imes 31}{6} = 1240$. Compute $1240 \mod 13$: $13 imes 95 = 1235$, so $1240 - 1235 = 5$. The remainder is $oxed{5}$.Question: A science educator is designing a lesson on probability using a digital simulation that generates random 6-digit numbers using only the digits 1, 2, and 3. What is the probability that a randomly generated number has exactly two consecutive digits that are the same?

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