How Many Ants?

At least a dozen ants are marching through my kitchen! If the ants walk in rows of 7, 11, or 13, there are 2 ants left over, while in rows of 10, there are 6 left over. What is the smallest number of ants there could be satisfying the above criterion?

Answer:

One thing to notice is that 7 * 11 * 13 = 1001, so the number can be (and must be) 2 more than a multiple of 1001. This gives 1003, 2004, 3005, 4006. The last one is the first one that's 6 more than a multiple of 10. Therefore there were 4006 ants.