Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Digits and Rationals.
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.
But try the problem first...
Answer: is 660.
AIME I, 1992, Question 5
Elementary Number Theory by David Burton
numbers relatively prime to 999 gives us the numerators