2017 AMC 8 Problems/Problem 19

Problem 19

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

Factoring a $98!$ , we have $98!(10,000)$ . So we have that $98//5 + 98//25 = 19 + 3 = 22$ factors of $5$ . Now $10,000$ has $4$ factors of $5$ , so there are $22 + 4 = 26$ , or $D$ .

2017 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
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All AJHSME/AMC 8 Problems and Solutions

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2017 AMC 8 Problems/Problem 19

Problem 19

Solution

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