Example 9 $3^{8833010}-59049$ can be divided by 24010000.
Given $24010000=245^{4} 7^{4}$ and Lemma 14, we have $\varphi(24010000)=2^{3} \times 5^{3} \times 4 \times 7^{3} \times 6=8232000$. Therefore, by Theorem 1 we have
$$3^{8232000} \equiv 1(\bmod 24010000)$$

By (28) we have
$$3^{8232010}-59049 \equiv 3^{10}-59049(\bmod 24010000)$$

By $3^{10}=59049$ and (29) we know that Example 9 holds.