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May 17, 2020

Positive divisor | AIME I, 1988 | Question 5

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1988 based on Positive divisor.

Positive Divisors- AIME I, 1988


Let \(\frac{m}{n}\) in lowest term, be the probability that a randomly chosen positive divisor of \(10^{99}\) is an integer multiple of \(10^{88}\). Find m+n.

  • is 107
  • is 634
  • is 840
  • cannot be determined from the given information

Key Concepts


Integers

DIvisors

Algebra

Check the Answer


Answer: is 634.

AIME I, 1988, Question 5

Elementary Number Theory by David Burton

Try with Hints


First hint

\(10^{99}=2^{99}5^{99}\)

or, (99+1)(99+1)=10000 factors

Second Hint

those factors divisible by \(10^{88}\)

are bijection to number of factors \(10{11}=2^{11}5^{11}\) has, which is (11+1)(11+1)=144

Final Step

one required probability =\(\frac{m}{n}=\frac{144}{10000}=\frac{9}{625}\)

m+n=634.

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