Number theory · Covering by primes · Consecutive integers · Counting multiples · Parity · Extremal problems
We want to choose a set \(P\) of \(k\) primes and a set \(N\) of \(n\) consecutive positive integers in such a way that every number \(a \in N\) is divisible by at least one prime \(p \in P\).
(a) Determine the largest possible \(n\) when \(k = 3\).
(b) Determine the smallest possible \(k\) when \(n = 13\).
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2025, 1. letnik, category A, problem B3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source