# Lagrange Interpolation?

Find the sum of all composite integers $$k$$ such that for any $$a_1,...,a_m \in \mathbb{Z}$$ which when taken $$\pmod k$$ are distinct, then $$∃ p(x)$$ a polynomial so that the following congruence relation

$$p(x) \equiv 0 \pmod k$$,

has exactly $$m$$ solutions where $$x \equiv a_1, a_2, ... , a_m \pmod k$$.

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