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 a factor of $10^{n}$ has a $2$ and a $5$ in its prime factorization, then that factor will end in a $0$ . Therefore, we have left to consider the case when the two factors have the $2$ s and the $5$ s separated, in other words whether $2^n$ or $5^n$ produces a 0 first.

$2^1 = 2 | 5^1 = 5$ $2^2 = 4 | 5 ^ 2 =25$ $2^3 = 8 | 5 ^3 = 125$

and so on, until,

$2^8 = 256$ | $5^8 = 390625$

We see that $5^8$ generates the first zero, so the answer is $\boxed{008}$ .

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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