MathDB

Let

S

be the set of all rational numbers that can be expressed as a repeating decimal in the form

0.\overline{abcd},

where at least one of the digits

a, b, c,

d

is nonzero. Let

N

be the number of distinct numerators when numbers in

S

are written as fractions in lowest terms. For example, both

4

and

410

are counted among the distinct numerators for numbers in

S

because

0.\overline{3636} = \frac{4}{11}

and

0.\overline{1230} = \frac{410}{3333}.

Find the remainder when

N

is divided by

1000.

Problem Statement