Question 44
Prove that there exist pairwise disjoint sets $A_1, A_2, \dots, A_{2014}$ whose union is the set of natural numbers and for which the following condition holds:
For arbitrary natural numbers $a$ and $b$, at least two of the numbers $a, b, \text{gcd}(a, b)$ belong to one of the sets $A_1, A_2, \dots, A_{2014}$.
Macedonian source version, translated in full:
Prove that there exist pairwise disjoint sets $A_1, A_2, \dots, A_{2014}$ whose union is the set of natural numbers and for which the following condition holds:
For arbitrary natural numbers $a$ and $b$, at least two of the numbers $a, b, \text{gcd}(a, b)$ belong to one of the sets $A_1, A_2, \dots, A_{2014}$.
For arbitrary natural numbers $a$ and $b$, at least two of the numbers $a, b, \text{gcd}(a, b)$ belong to one of the sets $A_1, A_2, \dots, A_{2014}$.
Macedonian source version, translated in full:
Prove that there exist pairwise disjoint sets $A_1, A_2, \dots, A_{2014}$ whose union is the set of natural numbers and for which the following condition holds:
For arbitrary natural numbers $a$ and $b$, at least two of the numbers $a, b, \text{gcd}(a, b)$ belong to one of the sets $A_1, A_2, \dots, A_{2014}$.