This problem is equivalent to asking how many positive integers not exceeding $1001$ are relatively prime to $1001$. This is a standard $\varphi$ function problem $$\varphi(1001)=1001\cdot\Big(1-\frac{1}{7}\Big)\Big(1-\frac{1}{11}\Big)\Big(1-\frac{1}{13}\Big)=\boxed{720}$$ Alternatively, the result can also be computed using the inclusion-exclusion principle: $$1001-\Big(\Big\lfloor{\frac{1001}{7}}\Big\rfloor+\Big\lfloor{\frac{1001}{11}}\Big\rfloor+\Big\lfloor{\frac{1001}{13}}\Big\rfloor\Big)+\Big(\Big\lfloor{\frac{1001}{7\times 11}}\Big\rfloor+\Big\lfloor{\frac{1001}{11\times 13}}\Big\rfloor+\Big\lfloor{\frac{1001}{13\times 1}}\Big\rfloor\Big)-\Big(\Big\lfloor{\frac{1001}{7\times 11\times 13}}\Big\rfloor\Big)=\boxed{720}$$ where $\lfloor{x}\rfloor$ return the biggest integer not exceeding $x$.