2015 AMC 10B Problems/Problem 23
Let be a positive integer greater than 4 such that the decimal representation of ends in zeros and the decimal representation of ends in zeros. Let denote the sum of the four least possible values of . What is the sum of the digits of ?
A trailing zero requires a factor of two and a factor of five. Since factors of two occur more often than factors of five, we can focus on the factors of five. We make a chart of how many trailing zeros factorials have:
We first look at the case when has zero and has zeros. If , has only zeros. But for , has zeros. Thus, and work.
Secondly, we look at the case when has zeros and has zeros. If , has only zeros. But for , has zeros. Thus, the smallest four values of that work are , which sum to . The sum of the digits of is
By Legendre's Formula and the information given, we have that .
We have as there is no way that if , would have times as many zeroes as .
First, let's plug in the number We get that , which is obviously not true. Hence,
After several attempts, we realize that the RHS needs to more "extra" zeroes than the LHS. Hence, is greater than a multiple of .
We find that the least are .
Solution 3 (Bashing)
We notice that for a to be at the end of a factorial, one multiple of five must be there. Therefore, it is intuitive to start at and work up. If you bash enough you get , , , and . Going any higher will give too many zeros, and then we can stop going higher. .
Let for some natural numbers , such that . Notice that . Thus For smaller , we temporarily let To minimize , we let , then Since , , the only integral value of is , from which we have .
Now we let and , then Since , .
If , then which is a contradiction.
Finally, the sum of the four smallest possible and .
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