1957 AHSME Problems/Problem 42
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Contents
Problem
If , where
and
is an integer, then the total number of possible distinct values for
is:
Solution 1
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
See Also
1957 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 41 |
Followed by Problem 43 | |
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