2014 AMC 10B Problems/Problem 17
Problem 17
What is the greatest power of that is a factor of
?
Solution
We begin by factoring the out. This leaves us with
.
We factor the difference of squares, leaving us with . We note that all even powers of 5 more than two end in ...
. Also, all odd powers of five more than 2 end in ...
. Thus,
would end in ...
and thus would contribute one power of two to the answer, but not more.
We can continue to factor as a difference of cubes, leaving us with
times an odd number.
ends in ...
, contributing two powers of two to the final result.
Adding these extra powers of two to the original
factored out, we obtain the final answer of
.
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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All AMC 10 Problems and Solutions |
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