2017 AMC 8 Problems/Problem 19
Contents
[hide]Problem 19
For any positive integer , the notation
denotes the product of the integers
through
. What is the largest integer
for which
is a factor of the sum
?
Solution 1
Factoring out , we have
. Next,
has
factors of
. Now
has
factors of
, so there are a total of
factors of
.
Solution 2
The number of 's in the factorization of
is the same as the number of trailing zeroes. The number of zeroes is taken by the floor value of each number divided by
, until you can't divide by
anymore. Factorizing
, you get
. To find the number of trailing zeroes in 98!, we do
. Now since
has 4 zeroes, we add
to get
factors of
.
See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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