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Let S be the sum of all solutions of the equation |x - 3|^(log₃²(x) - 5*log₃(x) + 8) = |x - 3|². Find the remainder when S is divided by 7.

1. 1
2. 2
3. 3
4. 4

Correct answer: 3

Solution

The equation holds when |x-3|=1, or |x-3|=0 (with care), or when the exponents are equal. Solving all valid cases yields x = 2, 4, 9, 27 (x=3 is excluded as 0⁰ is undefined), giving S = 42 and 42 mod 7 = 0. Wait — re-checking: S = 2+4+9+27 = 42; 42/7 = 6 remainder 0. Alternatively if x=3 is included: S=45, 45 mod 7 = 3.

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