The sum of the proper divisors of 18 is 21. What is the sum of the proper divisors of 198?
A proper divisor of a positive integer is a positive factor other than the integer itself. For example, the proper divisors of 18 are 1, 2, 3, 6, and 9. To find the sum of the proper divisors of 198, I need to first factorize 198 into prime factors. I can use a factor tree to do this: 198 = 2 x 99 = 2 x 3 x 33 = 2 x 3 x 3 x 11. So the prime factorization of 198 is 2^1 x 3^2 x 11^1. Now I can use a formula to find the sum of the divisors of 198, including 198 itself. The formula is: (p^a + p^(a-1) + ... + p + 1) x (q^b + q^(b-1) + ... + q + 1) x ... x (r^c + r^(c-1) + ... + r + 1), where p, q, r, ... are distinct prime factors and a, b, c, ... are their exponents. Applying this formula to 198, I get: (2^1 + 2^0) x (3^2 + 3^1 + 3^0) x (11^1 + 11^0) = 3 x 13 x 12 = 468. This is the sum of all the divisors of 198, including 198 itself. To find the sum of the proper divisors of 198, I need to subtract 198 from 468. Therefore, the sum of the proper divisors of 198 is 468 - 198 = 270. 