Problem:
Jim starts with a positive integer and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with , then his sequence contains numbers:
Let be the smallest number for which Jim’s sequence has numbers. What is the units digit of ?
Answer Choices:
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E.
Solution:
Let the sequence be . For for some such that . To minimize the value of , construct the sequence in reverse order and choose the smallest possible value of for each . The terms in reverse order are , , , and , which has the unit digit .
The problems on this page are the property of the MAA's American Mathematics Competitions