# 2017 AMC 8 Problems/Problem 19

## 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 $98!+99!+100!$ ?

$\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

## 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

## Solution 2

Also, keep in mind that the number of $5$’s in $98! (10,000)$ is the same as the number of trailing zeros. The number of zeros is $98!$, which means we need pairs of $5$’s and $2$’s; we know there will be many more $2$’s, so we seek to find the number of $5$’s in $98!$, which the solution tells us. And, that is $22$ factors of $5$. $10,000$ has $4$ trailing zeros, so it has $4$ factors of $5$ and $22 + 4 = \boxed{26}$.

~CHECKMATE2021

## Solution 3

We can first factor a $98!$ out of the $98! + 99! + 100!$ to get $98! ( 1 + 99 + 100*99 ),$ Simplify to get $98! (10,000)$.

Let's first find how many factors of $5 10,000$ has. $10,000$ is $(2*5)^4$ because $10,000$ is $(10)^4$. After we remove the brackets, we get $2^4$, and $5^4$. We only care about the latter (second one), because the problem only ask's for the power of $5$. We get $4$

Next, we can look at the multiples of 5 in $98!$. $98/5 = 19$ so there is 19 multiples of 5. We get $19$

But we cannot forget the multiples of $5$ with $2$ fives in it. Multiples of $25$. How many multiples of $25$ are between $1$ and $98$? $3$. $25,50,75,$ and that's it. We get $3$

Finally, we add all of the numbers (powers of $5$) up. That is $4 + 19 + 3$, which is just $26$

So the answer is $26$. Which is answer choice D $\boxed{\textbf{(D)}\ 26}$.

~CHECKMATE2021

## Video Solution (CREATIVE THINKING + ANALYSIS!!!)

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