An eccentric baseball card collector wants to distribute her collection among her descendants. If she divided her cards among her 17 great-great-grandchildren, there would be 3 cards left over. If she divided her cards among her 16 great-grandchildren, there would be 10 cards left over. If she divided her cards among her 11 grandchildren, there would be 4 cards left over. If she divided her cards among her 7 children, there would be no cards left over.
What is the smallest possible number of cards in her collection?
n mod 17 = 3, n mod 16 = 10, n mod 11 = 4, n mod 7 = 0, solve for n.
Since n mod 7=0, then the smallest number that satisfies all the conditions must be a whole multiple of 7 with no remainder. By simple iteration, it turns out that the multiple of 7 that satisfies all the conditions is =406. Therefore, the smallest number of cards in her collection would have to be:
7 x 406 =2,842 cards.
2,842 / 17 =167 with a remainder of 3, and:
2,842 / 16 =177 with a remainder of 10, and:
2,842 / 11 =258 with a remainder of 4, and:
2,842 / 7 =406 with a remainder of 0