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.

	1	2	3	4	5	6	7	8	9	10
Powers of $2$ :	$2$	$4$	$8$	$16$	$32$	$64$	$128$	$256$	$512$	$1\boxed{0}24$
Powers of $5$ :	$5$	$25$	$125$	$625$	$3125$	$15625$	$78125$	$39\boxed{0}265$

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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