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Home » So the square roots are \( 1 - i \) and \( -1 + i \) - Veritas Home Health

So the square roots are \( 1 - i \) and \( -1 + i \) - Veritas Home Health

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 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 $.

📌 We now find a two-digit value of $ x $:

📌 35m + 8 \geq 10 \Rightarrow m \geq 0, \quad 35m + 8 \leq 99 \Rightarrow m \leq 2

📌 Try $ m = 0 $: $ x = 8 $ (not two-digit)

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