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
have 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