Problem:
Consider the following operation. Given a positive integer , if is a multiple of 3 , then you replace by . If is not a multiple of 3 , then you replace by . Then continue this process. For example, beginning with , this procedure gives . Suppose you start with . What value results if you perform this operation exactly 100 times?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The first several iterations give
The values will then continue to cycle through . Note that the value 20 occurs after operations; the value 30 occurs after operations; and the value 10 occurs after operations. Because 100 has remainder 1 when divided by 3 , the value after 100 operations is .
The problems on this page are the property of the MAA's American Mathematics Competitions