2014 AMC 10B Problems/Problem 17
Contents
[hide]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
more than two end in ...
. Also, all odd powers of five more than
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
end in
, so the two powers of
will end with
. Adding
will make it end in
. 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
only. Still that's
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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.