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purdue boilermakers football vs notre dame fighting irish football stats - Decision Point

📅 May 6, 2026 👤 scraface

May 06, 2026

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📌 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:

📌 So the smallest integer $ m = 11 $, then:

📌 First, divide $ 98765 $ by $ 12345 $:

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