A positive integer X is defined as a bold number if precisely 8 of its distinct
divisors sum to 3240. The total number of divisors of X may or may not be equal to 8.
Determine the minimum positive bold number.
(In reply to
re: solution by Kenny M)
In order for X to have 8 divisors that sum to 3240, that means the
average of the 8 largest divisors must be at least 405. That means our
8 largest divisors are, at a minimum, (401, 402, 403, 404, 406, 407,
408, 409), as they must all be distinct divisors.
Using a computer program to find the sum of the 8 largest divisors, we
can find that 1458 (2 x 3^6) is the maximum value of X that could
possibly be the minimum bold number.
Going through and finding all values between 409 and 1458 such that the
8 largest divisors sum to at least 3240, gets us the list of all
possible minimum values of X.
That gets us 19 values to check: (1260, 1296, 1320, 1332, 1344, 1350,
1368, 1380, 1386, 1392, 1400, 1404, 1410, 1416, 1428, 1440, 1452, 1456,
1458)
Checking the above list of values does in fact confirm that 1260 works, and is the minimum positive value that can be a bold number.

Posted by Justin
on 20091212 19:45:08 