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)
 n = (a^2)/2
 n = (b^3)/3
 n = (c^5)/5
implies that n is divisible by 2, 3, and 5. The smallest n should only be divisible by these three factors or else it's not smallest. So let n=(2^r)(3^s)(5^t)
The first equation implies a^2=2n=(2^(r+1))(3^s)(5^t), thus
2r+1, 2s, 2t
Similarly, b^3=(2^r)(3^(s+1))(5^t) implies
3r, 3s+1, 3t
and c^5=(2^r)(3^s)(5^(t+1)) implies
5r, 5s, 5t+1
Combining them, we get
2r+1, 3r, 5r
2s, 3s+1, 5s
2t, 3t, 5t+1
Smallest n corresponds to the smallest r, s, t. And the smallest values are r=15, s=20, t=24
so n = (2^15)(3^20)(5^24)
that's pretty big.

Posted by Bon
on 20041127 18:29:31 