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?
A calculator may be helpful for this problem.
The number of cards in her collection is divisible by 7.
The number of cards in her collection must also be even - consider the 16 great-grandchildren, with x being the number of cards each child would receive. The number of cards in the collection equals 16x +10, or 2(8x+5), indicating that it is divisible by 2.
Thus, the number of cards in her collection is divisible by 7x2 = 14
At this point, you could begin investigating multiples of 14, to see when you first find one that yields 3 integer answers for the 3 cases. This could be done with a calculator, or more quickly with an Excel spreadsheet.
To find the smallest possible number of cards in her collection, we need to find the least common multiple (LCM) of the numbers 17, 16, 11, and 7 since those are the numbers of descendants in each generation.
Step 1: Find the LCM of 17 and 16.
The LCM of 17 and 16 is 272 because 17*16 = 272.
Step 2: Find the LCM of 272 and 11.
The LCM of 272 and 11 is 2,992 because 272*11 = 2,992.
Step 3: Find the LCM of 2,992 and 7.
The LCM of 2,992 and 7 is 20,944 because 2,992*7 = 20,944.
Thus, the smallest possible number of cards in her collection is 20,944.