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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


Answer: is 660.

AIME I, 1992, Question 5

Elementary Number Theory by David Burton

Try with Hints


Let x=0.abcabcabcabc…..

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

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

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

numbers relatively prime to 999 gives us the numerators

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

=660.

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