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best it's always sunny episodes - Parker Core Knowledge

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Solution: We seek the smallest three-digit number divisible by both $12$ and $18$, i.e., the least common multiple of $12$ and $18$, and then ensure it is the smallest three-digit such number.

📌 Now find the smallest three-digit multiple of $36$. The smallest three-digit number is $100$.

📌 Thus, every factor pair $(a,b)$ with $ab = 2025$ gives a solution, provided $a$ and $b$ are both odd (which they are), and we must consider both positive and negative factor pairs, since $a, b$ can be any integers (positive or negative) such that $ab = 2025$.

📌 Each negative pair: $a < 0$, $b < 0$, $ab = 2025$, so also 15 pairs.

📌 Each such pair gives a unique $(x,y)$:

📌 We must check for duplicate solutions. But each factor pair $(a,b)$ gives a distinct $(x,y)$, because $a$ and $b$ determine $x$ and $y$ uniquely.

📌 For example, $(a,b)$ and $(b,a)$ would give different $x$ unless $a = b$. But since $a$ and $b$ are interchangeable via the identity, but each ordered pair is counted once. However, in our setup, each ordered pair $(a,b)$ such that $ab = 2025$ is considered. But since $x$ and $y$ depend on $a$ and $b$ linearly, and $a$ ranges over all divisors, including both positions, we must avoid double-counting? No — each ordered pair $(a,b)$ with $ab = 2025$ is distinct and gives one solution.

📌 Each gives a unique lattice point. For instance:

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