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Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2000 based on Least Positive Integer.

## Least Positive Integer Problem – AIME I, 2000

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.

• is 107
• is 8
• is 840
• cannot be determined from the given information

### Key Concepts

Product

Least positive integer

Integers

But try the problem first…

Source

AIME I, 2000, Question 1

Elementary Number Theory by Sierpinsky

## Try with Hints

First hint

$10^{n}$ has factor 2 and 5

Second Hint

for n=1 $2^{1}$=2 $5^{1}$=5

for n=2 $2^{2}$=4 $5^{2}=25$

for n=3 $2^{3}$=8 $5^{3}=125$

……..

Final Step

for n=8 $2^{8}$=256 $5^{8}=390625$

here $5^{8}$ contains the zero then n=8.