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Home » \frac36288024 = 15120 - Get link 4share

\frac36288024 = 15120 - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 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:

📌 For $(1,25)$: $k - y = 1$, $k + y = 25$ → $k = 13$, $y = 12$ → $x = 26$.

📌 For $(5,5)$: $k - y = 5$, $k + y = 5$ → $k = 5$, $y = 0$ → $x = 10$.

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