Therefore, S(1800) = 19 * 7 * 31 = 4123.

Explanation: The exact number of running time depends on the computational model. When analyzing arithmetic with large numbers, we usually count either bit operations or arithmetic operations of size O(logn) (where n is the input size). Now, given the factorization of a number n, then the sum of divisors can be computed in polynomial time.

Therefore, S(1800) = 4 * 255 * 31 = 31620.

Explanation: To find the number of odd factors (which includes 1), we can exclude any power of 2 and do the same. So, for 456, we have (3 + 1)(1 + 1) = 8 odd positive factors.

Explanation: To find the number of even factors, we can multiply the number of even factors by the power of 2. For 3200, we have (5 + 1)(1 + 1)(2) = 24 even factors.

Explanation: The prime factorization of 1872 is 13 * 32 * 24 and S(24) = 1 + 2 + 4 + 8 + 16 = 31, S(52) = 1 + 5 + 25 = 31. Therefore, S(1872) = 31 * 31 * 13 = 12493.

Explanation: To find the number of odd factors, we can exclude any power of 2 and do the same. So, for 6500, we have (5 + 1)(1 + 1) = 6 * 2 = 12 odd positive factors.

Explanation: To find the number of even factors, we can multiply the number of even factors by the power of 2. For 5065, we have (3 + 1)(1 + 1)(2) = 4 * 2 * 2 = 16 even factors.

Therefore, S(1800) = 6 * 7 * 431 = 18102.

Explanation: To find the number of odd factors, we can exclude any power of 2 and do the same. So, for 6500, we have (5 + 1)(3 + 1)(1 + 1) = 6 * 4 * 2 = 48 odd positive factors.