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Divisibility with options

Source: Science ON 2021 Juniors/1

March 14, 2021
number theory

Problem Statement

Let a,p,q∈Z≥1a,p,q\in \mathbb{Z}_{\ge 1} be such that aa is a perfect square, a=pqa=pq and 2021 ∣ p3+q3+p2q+pq2.2021~|~p^3+q^3+p^2q+pq^2. Prove that 20212021 divides a\sqrt a.\\ \\ (Cosmin Gavrilă)
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