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Home » d^2 = 25 \\ - Silent Sales Machine

d^2 = 25 \\ - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 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 $x^2 - 4y^2 = 100$?

📌 Solution: Rewrite as $x^2 = 4y^2 + 100$. Let $x = 2k$, then $4k^2 = 4y^2 + 100 \implies k^2 - y^2 = 25$. Factor as $(k - y)(k + y) = 25$. The factor pairs of 25 are $(1,25), (5,5), (-1,-25), (-5,-5)$. Solving:

🔥 Popular Posts

eq b $, and $ |a| = |b| = 1 $. Let $ b = \overline{a} $ (since $ |a| =...
eq \overline{a} $ unless $ a $ is real, which contradicts $ a
eq b $. Instead, note that $ rac{a + b}{a - b} + rac{a - b}{a + b} =...
S = rac{2(e^{i heta} + e^{-i heta} + e^{i\phi} + e^{-i\phi})}{e^{i he...
This simplifies to $ rac{2(2\cos heta + 2\cos\phi)}{2i(\sin heta - \s...
S = rac{(a + b)^2 + (a - b)^2}{a^2 - b^2} = rac{2a^2 + 2b^2}{a^2 - b...
But $ a^2 - b^2 = (a - b)(a + b) $, and $ a^2 + b^2 = (a + b)^2 - 2ab...
S = rac{2(1 + (-1))}{1 - (-1)} = 0.
But earlier steps suggest $ S = rac{2(a^2 + b^2)}{a^2 - b^2} $. For $...
S = rac{2(e^{2i heta} + e^{2i\phi})}{e^{2i heta} - e^{2i\phi}} = 2 \c...

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