Difference between revisions of "2017 AMC 8 Problems/Problem 19"
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==Problem== | ==Problem== | ||
| − | For any positive integer <math>M</math>, the notation M! denotes the product of the integers 1 through | + | For any positive integer <math>M</math>, the notation <math>M!</math> denotes the product of the integers 1 through |
M. 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> ? | M. 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> ? | ||
<math>\textbf{(A) }23 \qquad \textbf{(B) }24 \qquad \textbf{(C) }25 \qquad \textbf{(D) }26 \qquad \textbf{(E) }327</math> | <math>\textbf{(A) }23 \qquad \textbf{(B) }24 \qquad \textbf{(C) }25 \qquad \textbf{(D) }26 \qquad \textbf{(E) }327</math> | ||
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==Solution 1== | ==Solution 1== | ||
Revision as of 03:18, 13 October 2024
Problem
For any positive integer 
, the notation 
denotes the product of the integers 1 through
M. 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 .The 
is because of all the multiples of 
. Now, 
has 
factors of , so there are a total of 
factors of .
~CHECKMATE2021
Note: Can you say what formula this uses? most AMC 8 test takers won't know it.
Video Solution (Omega Learn)
https://www.youtube.com/watch?v=HISL2-N5NVg&t=817s
~ GeometryMystery
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. 
