When x²⁰⁰⁷ is divided by (x² - 5x + 6), the remainder polynomial is R(x). Given that R(0) can be written as a*b*(a^c - b^c) where a, b, c are positive integers, find the value of (a + b + c - 2002).

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3. Since x² - 5x + 6 = (x-2)(x-3), the remainder R(x) = px + q is linear. By remainder theorem: R(2) = 2²⁰⁰⁷ and R(3) = 3²⁰⁰⁷. Solving: p = 3²⁰⁰⁷ - 2²⁰⁰⁷, q = 3*2²⁰⁰⁷ - 2*3²⁰⁰⁷ = 2*3*(2²⁰⁰⁶ - 3²⁰⁰⁶). So R(0) = 2*3*(2²⁰⁰⁶ - 3²⁰⁰⁶) giving a=2, b=3, c=2006. Then a+b+c-2002 = 2+3+2006-2002 = 9. Wait, rechecking: 2+3+2006 = 2011, 2011-2002 = 9.

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