Question

The set of positive integers is partitioned into several arithmetical progressions. Is it true that in at least one of them the first term is divisible by the common difference. (Enter $1$ if true or $0$ if false)

Answer 1

Let $a_i,r_i,i=1,\mathrm{2...},n$ be the first terms and the common differences of the progressions. Consider the number $r=r_1{r}_2...r_n$. This number must belong to one of the progressions, thus, for some i and k we have $r=r_1{r}_2...r_n=a_i+k{r}_i$. It follows that $a_i$ is divisible by $r=a_i$