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Home » You Won’t Believe This Cat’s Staring—It’s haunting! 😲 #CatStaringMeme Madness! - AMAZONAWS

You Won’t Believe This Cat’s Staring—It’s haunting! 😲 #CatStaringMeme Madness! - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 Thus, the remainder is $0$.

📌 Therefore, the remainder is $oxed{0}$.

📌 Question: What is the largest possible value of $\gcd(a,b)$ if the sum of two positive integers $a$ and $b$ is $2024$ and both are even?

📌 Solution: Let $a + b = 2024$, and suppose $d = \gcd(a,b)$. Since $d \mid a$ and $d \mid b$, then $d \mid (a + b) = 2024$. So $d$ is a divisor of $2024$.

📌 Additionally, since both $a$ and $b$ are even, $d$ must be even (because any common divisor of two even numbers is even).

📌 To maximize $d$, we take the largest even divisor of $2024$ such that $a$ and $b$ are positive integers.

📌 Factor $2024$:

📌 = 2^3 imes 11 imes 23.

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$y = 0$: $25x^2 = 3600 \Rightarrow x^2 = 144$, $x = \pm 12$.
$y = \pm1$: $25x^2 = 144 + 3600 = 3744$, $x^2 = 3744/25 = 149.76$, not...
$y = \pm5$: $25x^2 = 144(25) + 3600 = 3600 + 3600 = 7200$, $x^2 = 288$...
Eventually, only solutions are $(\pm12, 0)$.
Thus, exactly 2 lattice points.
Question: A high-performance computing simulation requires a grid size...
Solution: We seek the smallest multiple of $\mathrm{lcm}(1,2,\dots,12)...
Factor each number from 1 to 12:
$2^3$ (from 8),
$3^2$ (from 9),

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