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Home » n^2 + 3n + 2 = (n + 1)(n + 2) - 500apps

n^2 + 3n + 2 = (n + 1)(n + 2) - 500apps

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 We apply the Chinese Remainder Theorem step by step.

📌 First, solve the first two congruences:

📌 Let \( k = 5a + 3 $. Substitute into the second:

📌 5a + 3 \equiv 4 \pmod{7} \implies 5a \equiv 1 \pmod{7}

📌 Multiply both sides by the modular inverse of 5 modulo 7. Since \( 5 \cdot 3 = 15 \equiv 1 \mod 7 \), the inverse is 3:

📌 a \equiv 3 \cdot 1 = 3 \pmod{7} \implies a = 7b + 3

📌 k = 5(7b + 3) + 3 = 35b + 15 + 3 = 35b + 18

📌 So \( k \equiv 18 \pmod{35} \)

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First, compute \( f(3) \).
f(3) = 3(3)^2 + 2(3) - 5 = 3 \times 9 + 6 - 5 = 27 + 6 - 5 = 28
Now, substitute \( f(3) = 28 \) into \( g(x) \):
g(28) = \sqrt{28 + 7} = \sqrt{35}
Thus, the value of \( g(f(3)) \) is \(\sqrt{35}\).
\boxed{\sqrt{35}}
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