The following problem is solved by using factors and multiples and features the strategies of guessing and checking and making an organized list.

Problem

A factory uses machines to sort cards into piles. On one occasion a machine operator obtained the following curious result.

When a box of cards was sorted into 7 equal groups, there were 6 cards left over, when the box of cards was sorted into 5 equal groups, there were 4 left over, and when it was sorted into 3 equal groups, there were 2 left.

If the machine cannot sort more than 200 cards at a time, how many cards were in the box?