2014 AMC 10B Problems/Problem 17
Contents
Problem
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
, and is divisible by
only. Still that's
powers of
.
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
.
Solution 3
Convert . We can factor out
to get that
. Using the adjusted Lifting The Exponent lemma (
for all even
and odd
), we get that the answer is
Solution 4
Factor out to get
. Since
, but
,
has 3 factors of 2. Hence
is the largest power of two which divides the given number
Solution 4
Like Solution 1, factor out to get
. Using general observation, we observe that for any positive integer 5^n (where n is an odd positive integer), it appears that the least even numbers directly above and below n in value must contain a maximum multiple of 4 and a maximum multiple of 2. Hence, the answer is 2^(1002+2+1) which is
Video Solution
~savannahsolver
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 |
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