Example 3 Determine the number of solutions to the following congruence equations:
(i) $x^{2} \equiv -1(\bmod 61)$;
(ii) $x^{2} \equiv 16(\bmod 51)$;
(iii) $x^{2} \equiv -2(\bmod 209)$;
(iv) $x^{2} \equiv -63(\bmod 187)$.
From Corollary 3, we know that the number of solutions to equation (i) is 2. The congruence equation (ii) is equivalent to the system of equations: \(x^{2} \equiv 1(\bmod 3), x^{2} \equiv-1(\bmod 17)\). Both of these equations have 2 solutions each, and by Theorem 1 of § 4, the number of solutions to the congruence equation (ii) is 4. The congruence equation (iii) is equivalent to the system of equations
$$x^{2} \equiv-2(\bmod 11), \quad x^{2} \equiv-2(\bmod 19),$$

From the table in Example 1, we know that both of these equations have 2 solutions each, and by Theorem 1 of § 4, the number of solutions to the congruence equation (iii) is 4. The congruence equation (iv) is equivalent to the system of equations
$$x^{2} \equiv 3(\bmod 11), x^{2} \equiv 5(\bmod 17)$$

From the table in Example 1, we know that the latter equation has no solutions, so the original equation has no solutions.