Problem:
Josh writes the numbers . He marks out 1 , skips the next number (2), marks out 3 , and continues skipping and marking out the next number to the end of his list. Then he goes back to the start of his list, marks out the first remaining number (2), skips the next number (4), marks out 6 , skips 8 , marks out 10 , and so on to the end. Josh continues in this manner until only one number remains. What is that number?
Answer Choices:
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B.
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E.
Solution:
In the first pass Josh marks out the odd numbers , leaving the multiples of . In the second pass Josh marks out , leaving the multiples of . Similarly, in the pass Josh marks out the numbers that are not multiples of , leaving the numbers that are multiples of . It follows that in the pass Josh marks out the numbers that are multiples of but not multiples of , namely 32 and 92 . This leaves 64 , the only number in his original list that is a multiple of . Thus the last number remaining is .
The problems on this page are the property of the MAA's American Mathematics Competitions