Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2000 based on Least Positive Integer.
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.
Product
Least positive integer
Integers
Answer: is 8.
AIME I, 2000, Question 1
Elementary Number Theory by Sierpinsky
\(10^{n}\) has factor 2 and 5
for n=1 \(2^{1}\)=2 \(5^{1}\)=5
for n=2 \(2^{2}\)=4 \(5^{2}=25\)
for n=3 \(2^{3}\)=8 \(5^{3}=125\)
........
for n=8 \(2^{8}\)=256 \(5^{8}=390625\)
here \(5^{8}\) contains the zero then n=8.
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