What is the largest power of 5 that divides the product of the first 100 positive integers?

June 29th, 2009

I need help on how to solve this problem, can you please explain how you did so I can understand. Power of 5 is like 5^x and remember, it's the LARGEST power of 5 that DIVIDES the PRODUCT of 1x2x3…98x99x100. Thanks!
The product of those integers is a multiple of 5 because I've found a pattern showing it would end with 0. I still don't have the answer though.

The only numbers that contribute to this power of 5 are those that are multiples of 5. There are 100/5 = 20 multiples of 5.

Also, multiples of 25 = 5^2 contribute another power of 5 apiece.
There are 100/25 = 4 multiples of 25.

So, the largest power of 5 dividing 100! is 20 + 4 = 24.

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2 Responses to “What is the largest power of 5 that divides the product of the first 100 positive integers?”

  1. Comment by Jacob
    June 30th, 2009

    no solution, 5^anything cant divide anything else unless it is a multipe of 5, if u multply 2,3,4,6,7,8,9 in those 100 #s, it is impossible
    References :

  2. Comment by kb
    June 30th, 2009

    The only numbers that contribute to this power of 5 are those that are multiples of 5. There are 100/5 = 20 multiples of 5.

    Also, multiples of 25 = 5^2 contribute another power of 5 apiece.
    There are 100/25 = 4 multiples of 25.

    So, the largest power of 5 dividing 100! is 20 + 4 = 24.
    References :