Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2010 based on Probability of divisors.

Probability of divisors – AIME I, 2010

Ramesh lists all the positive divisors of \(2010^{2}\), she then randomly selects two distinct divisors from this list. Let p be the probability that exactly one of the selected divisors is a perfect square. The probability p can be expressed in the form \(\frac{m}{n}\), where m and n are relatively prime positive integers. Find m+n.

  • is 107
  • is 250
  • is 840
  • cannot be determined from the given information

Key Concepts



Number Theory

Check the Answer

But try the problem first…

Answer: is 107.

Suggested Reading

AIME I, 2010, Question 1

Elementary Number Theory by Sierpinsky

Try with Hints

First hint


Second Hint

\((2+1)^{4}\) divisors, \(2^{4}\) are squares

Final Step

probability is \(\frac{2.2^{4}.(3^{4}-2^{4})}{3^{4}(3^{4}-1)}=\frac{26}{81}\) implies m+n=107

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