Nice work - the answer is A.
I'll post my solution as well, since I already wrote it:
Begin by finding the prime factorization of 1440, which is 1440 = 2×2×2×2×2×2×3×3×5,
So, our product k must contain at least five 2’s, two 3’s and one 5
Begin at 1 and keep adding additional numbers to the product k, until we meet this criterion.
 1×2×3×(2×2)×5×(2×3)×7×(2×2×2) (STOP here – we have our five 2’s, two 3’s and one 5)
We need only progress from 1 to 8 until we have all of the prime factors of 1440 in the product.
Brent Hanneson - Creator of GMATPrepNow.com
