Summary: For integers \(p\), \(b \ge 2\), suppose that \(D=\left\{ 0, 1,\dots, b-1\right\}\) is a consecutive digit set. It’s noted that the Cantor measure \(\mu_{pb,D}\) is spectral with a spectrum \[ \Lambda \left(pb, pD\right) =\left\{ \sum^{finite}_{j=0} \left(pb\right)^j d_j:d_j\in pD \right\}. \] By building the connection with number theory, we aim to explore the conditions of the integer \(\tau\) under which the scaling set \(\tau \Lambda \left(pb, pD\right)\) is also the spectrum of \(\mu_{pb,D}\). If so, we call \(\tau\) complete. In particular, for prime numbers \(\tau\), \(\tau_1\), \(\tau_2,\dots, \tau_m\) and \(\tau_i > pb-1\), we investigate the sufficient conditions that the power of \(\tau\) coprime to pb is complete and the power of \(\tau_1 \tau_2 \cdot \cdot \cdot \tau_m\) is complete. Furthermore, when an integer \(\tau\) coprime to \(b\) is incomplete while every proper divisor of it is complete, we call \(\tau\) primitive. So we obtain some properties and a criteria for the primitive number.