Problem:
Suppose there is a special key on a calculator that replaces the number currently displayed with the number given by the formula . For example, if the calculator is displaying and the special key is pressed, then the calculator will display since . Now suppose that the calculator is displaying . After the special key is pressed times in a row, the calculator will display
Answer Choices:
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Solution:
After the special key is pressed once, the calculator display reads since . If the key is pressed again, the calculator display reads since . If the key is pressed a third time, the calculator display reads , since . Thus pressing the special key three times returns to the original calculator display. The calculator display will continue to cycle through the three answers , and . Since is more than a multiple of , the calculator display will be .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions