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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Solution: A lattice point is a point $ (x, y) $ where both $ x $ and $ y $ are integers. The given equation is:

📌 Multiply both sides by $ 144 $ (the least common multiple of 16 and 9):

📌 We now search for integer solutions $ (x, y) $ satisfying this Diophantine equation. Rearranging:

📌 Try small integer values of $ y $ and solve for $ x $:

📌 Question: What is the smallest four-digit number that is divisible by $ 13 $ and leaves a remainder of $ 5 $ when divided by $ 7 $?

📌 Solution: We seek the smallest four-digit number $ N $ such that:

📌 So $ k = 7m + 2 $, and $ N = 13k = 13(7m + 2) = 91m + 26 $

📌 We want the smallest four-digit $ N $, so:

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