2014 AMC 10B Problems/Problem 17
Contents
Problem 17
What is the greatest power of that is a factor of
?
Solution 1
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 (Notice that the other number is
. The powers of 5 end in
, so the two powers of
will end with 0. Adding 1 will make it end in 1. Thus, this is an odd number).
ends in ...
, contributing two powers of two to the final result.
Or we can see that ends in 124, and is divisible by 2 only. Still that's 2 powers of 2.
Adding these extra powers of two to the original
factored out, we obtain the final answer of
.
Solution 2
First, we can write the expression in a more primitive form which will allow us to start factoring.
Now, we can factor out
. This leaves us with
. Call this number
. Thus, our final answer will be
, where
is the largest power of
that divides
. Now we can consider
, since
by the answer choices.
Note that
The powers of
cycle in
with a period of
. Thus,
This means that
is divisible by
but not
, so
and our answer is
.
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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