2017 AMC 8

Complete problem set with solutions and individual problem pages

Problem 19 Hard

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! ?

  • A.

    23

  • B.

    24

  • C.

    25

  • D.

    26

  • E.

    27

Answer:D

Factoring out 98!+99!+100!, we have 98! (1+99+99 \cdot 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.

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