2000 AIME I Problems/Problem 1

Problem

Find the least positive integer $n$ such that no matter how $10^{n}$ is expressed as the product of any two positive integers, at least one of these two integers contains the digit $0$ .

Solution

If there is a 2 and a 5 in one of the factors, then that factor will have a 0 in it. Therefore, the worst case scenario is where the 2's and the 5's are separated. Therefore, we need to find which $2^n$ or $5^n$ produces a 0 first.

Thinking back to our powers of 2, $2^{10}$ is the first power of 2 with a 0 in it.

For our powers of 5, after a little multiplication, we find that $5^8=390625$ is the first power of 5 with a 0 in it.

$\boxed{008}$

2000 AIME I (Problems • Answer Key • Resources)
Preceded by First Question	Followed by Problem 2
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2000 AIME I Problems/Problem 1

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