1957 AHSME Problems/Problem 42
Problem 42
If , where and is an integer, then the total number of possible distinct values for is:
Solution
We first use the fact that . Note that and , so and are periodic with periods at most 4. Therefore, it suffices to check for .
For , we have .
For , we have .
For , we have .
For , we have .
Hence, the answer is .
Solution 2
Notice that the powers of cycle in cycles of 4. So let's see if is periodic.
For : we have .
For : we have .
For : we have .
For : we have .
For : we have again. Well, it can be seen that cycles in periods of 4. Select .
~hastapasta