The number of muons in an air shower is a strong indicator of the mass number~$A$ of the primary cosmic-ray, increasing as a small power of it, $N_{\mu} \sim A^{(1-\beta)}$, where the exponent~$\beta$ is slightly less than~$1$. This behaviour can be explained in terms of the Heitler--Matthews model of hadronic air showers. In this paper, we present a method for calculating $\beta$ from the Heitler--Matthews model. The method has been successfully verified with a series of simulated events corresponding to events observed by the Pierre Auger Observatory at $10^{19}$ eV\@. To follow real measurements of the mass composition at this energy, the generated sample consists of certain fractions of events produced with p, He, N and Fe primary particles. Since hadronic interactions at the highest energies can differ from those observed at energies reached by terrestrial accelerators, we generate a mock dataset with $\beta =0.92$ (the canonical value) and $\beta =0.96$ (a more exotic scenario). The method can be applied to measured events to determine the muon signal for each primary particle as well as the muon scaling factor and the $\beta$ exponent. Determining the $\beta$ exponent can effectively constrain parameters that govern hadronic interactions and help resolve the so-called muon problem, where hadronic interaction models predict too few muons relative to the observations. In this paper, through a simulation study, we lay foundations for future analyses of measured data from the Pierre Auger Observatory.