MathDB
Sets of multiples

Source:

November 25, 2019
GCDLCMDivisibilitynumber theory

Problem Statement

For any a∈Z≥0 a\in\mathbb{Z}_{\ge 0} make the notation aZ≥0={an∣n∈Z≥0}. a\mathbb{Z}_{\ge 0} =\{ an| n\in\mathbb{Z}_{\ge 0} \} . Prove that the following relations are equivalent:
(1)aZ≥0∖bZ≥0⊂cZ≥0∖dZ≥0 \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} (2)b∣a or (c∣a and lcm(a,b)∣lcm(a,d)) \text{(2)} b|a\text{ or } (c|a\text{ and } \text{lcm} (a,b) |\text{lcm} (a,d))
Marin Tolosi and Cosmin Nitu
Back to ProblemsView on AoPS