# Find The Largest N

Algebra Level 5

Consider the function $f(x) = \dfrac{(x-1)^{1093^{2014}} - x^{1093^{2014}} + 1}{1093(x-1)}.$ Find the last three digits of the largest positive integer $$N$$ such that there exist two polynomials $$a(x)$$ and $$b(x)$$ both having integer coefficients and an integer $$c$$ (not necessarily positive) satisfying $f(x) = (x-c)^N a(x) + b(x).$

Details and assumptions

• You might use the fact that $$1093$$ is a prime and $$2^{1092}-1$$ is a multiple of $$1093^2$$ (such primes are called Wieferich primes).

• This problem is not original.

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