HMMT 二月 2002 · 代数 · 第 6 题

英文解析

6. Find the sum of the even positive divisors of 1000.
   Solution: 2184 . Notice that 2 k is a divisor of 1000 iff k is a divisor of 500, so we need 1
   only find the sum of the divisors of 500 and multiply by 2. This can be done by enumerating
   2 3 2
   the divisors individually, or simply by using the formula: σ (2 · 5 ) = (1 + 2 + 2 )(1 + 5 +
   2 3
   5 + 5 ) = 1092, and then doubling gives 2184. Alternate Solution: The sum of all the
   2 3 2 3
   divisors of 1000 is (1 + 2 + 2 + 2 )(1 + 5 + 5 + 5 ) = 2340. The odd divisors of 1000 are
   2 3
   simply the divisors of 125, whose sum is 1 + 5 + 5 + 5 = 156; subtracting this from 2340,
   we are left with the sum of the even divisors of 1000, which is 2184.