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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