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AIME I Algebra Arithmetic Math Olympiad USA Math Olympiad

Digits and Rationals | AIME I, 1992 | Question 5

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Digits and Rationals.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Digits and Rationals.

Digits and Rationals – AIME I, 1992


Let S be the set of all rational numbers r, 0<r<1, that have a repeating decimal expression in the form 0.abcabcabcabc…. where the digits a,b and c are not necessarily distinct. To write the elements of S as fractions in lowest terms find number of different numerators required.

  • is 107
  • is 660
  • is 840
  • cannot be determined from the given information

Key Concepts


Integers

Digits

Prime

Check the Answer


But try the problem first…

Answer: is 660.

Source
Suggested Reading

AIME I, 1992, Question 5

Elementary Number Theory by David Burton

Try with Hints


First hint

Let x=0.abcabcabcabc…..

\(\Rightarrow 1000x=abc.\overline{abc}\)

\(\Rightarrow 999x=1000x-x=abc\)

\(\Rightarrow x=\frac{abc}{999}\)

Second Hint

numbers relatively prime to 999 gives us the numerators

\(\Rightarrow 999(1-\frac{1}{3})(1-\frac{1}{111})\)=660

Final Step

=660.

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