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Home » \(1 \cdot 2 \cdot 3 = 6\), which is not divisible by 12. - Silent Sales Machine

\(1 \cdot 2 \cdot 3 = 6\), which is not divisible by 12. - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 \frac{16}{56} = \frac{2}{7}

📌 The probability that no two selected stations are adjacent is $\boxed{\frac{2}{7}}$.Question: What is the largest possible value of $\gcd(a,b)$ if the sum of two positive integers $a$ and $b$ is 2025, representing the years of two tectonic shifts?

📌 Sكية: The greatest common divisor of $a$ and $b$ must divide their sum, which is 2025. To maximize $\gcd(a,b)$, we find the largest proper divisor of 2025. Factoring 2025: $2025 = 3^4 \times 5^2$. The largest proper divisor is $2025 / 3 = 675$. Thus, the maximum $\gcd(a,b)$ is $\boxed{675}$.

📌 Question: What is the remainder when the total number of quantum measurements, $11071 + 11073 + 11075 + 11077$, is divided by 13?

📌 Solution: Compute the sum: $11071 + 11073 = 22144$, $11075 + 11077 = 22152$, total $22144 + 22152 = 44296$. Divide 44296 by 13. Since $13 \times 3407 = 44291$, the remainder is $44296 - 44291 = 5$. The remainder is $\boxed{5}$.

📌 Question: What two-digit positive integer is one more than a multiple of 7 and one more than a multiple of 11?

📌 Solution: Let $x \equiv 1 \pmod{7}$ and $x \equiv 1 \pmod{11}$. By the Chinese Remainder Theorem, $x \equiv 1 \pmod{77}$. The two-digit numbers satisfying this are $1 + 77 = 78$. Thus, the answer is $\boxed{78}$.

📌 Question: How many lattice points lie on the hyperbola $x^2 - 4y^2 = 100$?

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