Difference between revisions of "2017 AMC 8 Problems/Problem 19"
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For any positive integer <math>M</math>, the notation <math>M!</math> denotes the product of the integers <math>1</math> through <math>M</math>. What is the largest integer <math>n</math> for which <math>5^n</math> is a factor of the sum <math>98!+99!+100!</math> ? | For any positive integer <math>M</math>, the notation <math>M!</math> denotes the product of the integers <math>1</math> through <math>M</math>. What is the largest integer <math>n</math> for which <math>5^n</math> is a factor of the sum <math>98!+99!+100!</math> ? | ||
Revision as of 14:08, 18 January 2021
Problem
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
which is
Next,
has
factors of
. The
is because of all the multiples 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
.
Solution 3
We can rewrite the expression as .
The exponent of
in
is
. Onto the
part.
Remember that is the product of the integers from 1 to 98. Among these, there are multiples of
and multiples of
.
The number of multiples of
below or equal to
is
(try to see why), which is
. Every such number has a factor of
, so they contribute
to the total each. So these numbers contribute
to the exponent of
in
.
However, we forgot about multiples of . Using similar logic, we have
, so there are 3 multiples of 25 in this range. Multiples of 25 contribute 2 to the total each. We already counted 1 of 2 contributions while counting multiples of 5. So we need to add another 1 to the exponent of 5 for every multiple of 25. So these numbers contribute 3 to the exponent of 5 in
.
We thus have that the exponent of in
is
, and so our answer is
.
~Math4Life2020
See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.