2019 AMC 10B Problems/Problem 14
Contents
Problem
The base-ten representation for is
, where
,
, and
denote digits that are not given. What is
?
Solution 1
We can figure out by noticing that
will end with
zeroes, as there are three
's in its prime factorization. Next we use the fact that
is a multiple of both
and
. Since their divisibility rules gives us that
is congruent to
mod
and that
is congruent to
mod
. By inspection, we see that
is a valid solution. Therefore the answer is
, which is (C).
Solution 2
We can manually calculate 19!. If we prime factorize 19!, it becomes . This looks complicated, but we can use elimination methods to make it simpler.
, and
. If we put these aside for a moment, we have
.
, and
. We have the 2's and 3's out of the way, and then we have
. Now if we multiply all the values calculated, we get
. Thus
, and the answer
, thus (C).
See Also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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