Difference between revisions of "2017 AMC 8 Problems/Problem 19"
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+ | ==Problem== | ||
+ | For any positive integer <math>M</math>, the notation M! 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> ? | ||
+ | |||
+ | <math>\textbf{(A) }23 \qquad \textbf{(B) }24 \qquad \textbf{(C) }25 \qquad \textbf{(D) }26 \qquad \textbf{(E) }327</math> | ||
+ | |||
==Solution 1== | ==Solution 1== |
Revision as of 21:49, 4 July 2024
wertesryrtutyrudtu
Problem
For any positive integer , the notation M! 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. Also, can someone unvandalize this page?
Video Solution (CREATIVE THINKING + ANALYSIS!!!)
https:/90ijn bidxrfgv