The April/May, 2007 PoW

Let us assume we have a perfect analog watch. That is, one whose hands move continuously at the same rate. So at 3:00, the big hand is pointing to 12, and the little hand points to 3. At 3:15, the big hand is a quarter of the way around, pointing to 3, and the little hand will be pointing one quarter of the way between 3 and 4.

otter

Eventually the big hand will catch up to the little hand, and both will be pointing in the exact same direction. Find all such times.