Number theory · Primes · Perfect squares · Divisibility · Prime factorization
Find all pairs of prime numbers \(p\) and \(q\) for which the number
\[ 2p^{2}q + 45pq^{2} \]
is a perfect square.
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2008, 1. letnik, category A, problem 1. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source