What's the least positive integer,
n, having the following properties:
 n = (a^2)/2
 n = (b^3)/3
 n = (c^5)/5
(where a, b, and c are integers)
As we want N to be as small as possible, we insist that it is composed of powers of 2, 3 and 5. (Other factors simply make N larger without helping to fulfill any of the constraints. We have
N = 2^á 3^â 5^è Then the three eqalities give
* á is odd, â and è are even
* â is one more than a multiple of 3, á and è are multiples of 3
* è is one more than a multiple of 5, á and â are multiples of 5.
The smallest (á, â, è) fitting these conditions is (15,10,6). Hence
N = 2^15 * 3^10 * 5^6

Posted by FrankM
on 20080116 00:28:47 