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how many ounces is 1/4 cup - Decision Point

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 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?

📌 Solution: To solve this, we must count the number of 6-digit numbers using only the digits 1, 2, and 3 such that exactly one pair of consecutive digits are equal, and all other adjacent digits are different.

📌 Letâs denote the digits of the number as $ d_1, d_2, d_3, d_4, d_5, d_6 $, where each $ d_i \in \{1, 2, 3\} $.

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