Question

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: $98!+99!+100!$

$23$

$24$

$25$

$26$

$27$

Answer: D

Concepts:factorial Legendre’s Formula

Difficulty rating: 1640

Video solution:

Click to load, then click again to play

Written solution:

Factor out $98!$ : $\begin{aligned} 98!+99!+100! &=98!\bigl(1+99\\ &\qquad+100\cdot99\bigr)\\ &=98!\cdot100^2. \end{aligned}$

The exponent of $5$ in $98!$ is $\left\lfloor98/5\right\rfloor+\left\lfloor98/25\right\rfloor$ , which equals $19+3=22.$ The factor $100^2$ contributes four more factors of $5$ , so the total exponent is $22+4=26.$

Thus, D is the correct answer.

2017 AMC 8 Problem 19

Answer: D

Difficulty rating: 1640

Video solution:

Written solution:

Problem 19 in Other Years