Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2017 AMC 8 Problems/Problem 19: Difference between revisions

← Older editNewer edit →
Geometrymystery (talk | contribs)
editor
16 edits
Geometrymystery (talk | contribs)
editor
16 edits
Line 15: Line 15:
Note: Can you say what formula this uses? most AMC 8 test takers won't know it. Also, can someone unvandalize this page?
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!!!)==
==Video Solution (Omega Learn)==
https:/90ijn bidxrfgv
https://www.youtube.com/watch?v=HISL2-N5NVg&t=817s
 
~ GeometryMystery

Revision as of 20:57, 4 July 2024

wertesryrtutyrudtu

Problem

For any positive integer $M$, the notation M! denotes the product of the integers 1 through M. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98!+99!+100!$ ?

$\textbf{(A) }23 \qquad \textbf{(B) }24 \qquad \textbf{(C) }25 \qquad \textbf{(D) }26 \qquad \textbf{(E) }327$


Solution 1

Factoring out $98!+99!+100!$, we have $98! (1+99+99*100)$, which is $98! (10000)$. Next, $98!$ has $\left\lfloor\frac{98}{5}\right\rfloor + \left\lfloor\frac{98}{25}\right\rfloor = 19 + 3 = 22$ factors of $5$. The $19$ is because of all the multiples of $5$.The $3$ is because of all the multiples of $25$. Now, $10,000$ has $4$ factors of $5$, so there are a total of $22 + 4 = \boxed{\textbf{(D)}\ 26}$ factors of $5$.

~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 (Omega Learn)

https://www.youtube.com/watch?v=HISL2-N5NVg&t=817s

~ GeometryMystery

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_8_Problems/Problem_19&oldid=221690"