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Found problems: 51

2012 Bogdan Stan, 2
Tags: GCD , LCM , Divisibility , number theory
For any $ a\in\mathbb{Z}_{\ge 0} $ make the notation $ a\mathbb{Z}_{\ge 0} =\{ an| n\in\mathbb{Z}_{\ge 0} \} . $ Prove that the following relations are equivalent: $ \text{(1)} a\mathbb{Z}_{\ge 0} \setminus b\mathbb{Z}_{\ge 0}\subset c\mathbb{Z}_{\ge 0} \setminus d\mathbb{Z}_{\ge 0} $ $ \text{(2)} b|a\text{ or } (c|a\text{ and } \text{lcm} (a,b) |\text{lcm} (a,d)) $ [i]Marin Tolosi[/i] and [i]Cosmin Nitu[/i]