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Home » Substitute $ y = 10 - x $ into the second equation: - Ready Digital AB

Substitute $ y = 10 - x $ into the second equation: - Ready Digital AB

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 + 1 + 3 + 5 = 16 \equiv 0 \pmod{8}

📌 Question: What two-digit positive integer is one more than a multiple of $ 7 $ and leaves a remainder of $ 3 $ when divided by $ 5 $?

📌 Solution: Let $ x $ be the two-digit number. We are given:

📌 x \equiv 1 \pmod{7}, \quad x \equiv 3 \pmod{5}

📌 We solve this system using the Chinese Remainder Theorem.

📌 Let $ x = 7k + 1 $. Substitute into the second congruence:

📌 7k + 1 \equiv 3 \pmod{5} \Rightarrow 7k \equiv 2 \pmod{5} \Rightarrow 2k \equiv 2 \pmod{5} \Rightarrow k \equiv 1 \pmod{5}

📌 So $ k = 5m + 1 $, and $ x = 7(5m + 1) + 1 = 35m + 8 $.

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