2005 Alabama ARML TST Problems/Problem 13
Problem
There is one natural number with exactly 6 positive divisors, the sum of whose reciprocals is 2. Find that natural number.
Solution
Let the number be , and let
and
be primes. Therefore, one of the following is true:
For the first one, the sum of the reciprocals of the divisors of is therefore
. The smallest prime (2) makes that less than 2, and if
gets bigger, then that expression gets smaller, so there is absolutely no way that
. So the second case is true.
Therefore, . Now the only way that that is possible is when
. Solving for
, we get that
. Checking, the sum of the reciprocals of the divisors of
indeed sum to 2, and 28 does have 6 factors.