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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