RU-ALLRUSSIAN 1966 Problem 5

A rectangle ABCD is drawn on squared paper with its vertices at lattice points and its sides lying along the gridlines. $\mathrm{AD} = \mathrm{k}$ AB with $\mathrm{k}$ an integer. Prove that the number of shortest paths from A to C starting out along AD is $\mathrm{k}$ times the number starting out along AB.