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AIME I Algebra Arithmetic Functional Equations Math Olympiad USA Math Olympiad

Least Positive Integer Problem | AIME I, 2000 | Question 1

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

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

Check the Answer


But try the problem first…

Answer: is 8.

Source
Suggested Reading

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.

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