Školjka

For every real number $x$, let $\lVert x \rVert$ denote the distance between $x$ and the nearest integer. Prove that for every pair $(a, b)$ of positive integers there exist an odd prime $p$ and a positive integer $k$ satisfying $$ \left\lVert \frac{a}{p^k} \right\rVert + \ \left\lVert \frac{b}{p^k} \right\rVert + \ \left\lVert \frac{a + b}{p^k} \right\rVert = 1 \text{.} $$ \begin{flushright}\emph{(Hungary)}\end{flushright}