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 2009-12-12 19:45:08