ILL 1984 Problem 37

$(MOR 1)$ Denote by $[x]$ the greatest integer not exceeding $x$ . For all real $k > 1$ , define two sequences: $$a_n(k) = [nk]\text{ and } b_n(k) =\left[\frac{nk}{k - 1}\right]$$ If $A(k) = \{a_n(k) : n \in\mathbb{N}\}$ and $B(k) = \{b_n(k) : n \in \mathbb{N}\}$ , prove that $A(k)$ and $B(k)$ form a partition of $\mathbb{N}$ if and only if $k$ is irrational.