MathDB

By a pure repeating decimal (in base

10

), we mean a decimal

0.\overline{a_1\cdots a_k}

which repeats in blocks of

k

digits beginning at the decimal point. An example is

.243243243\cdots = \tfrac{9}{37}

. By a mixed repeating decimal we mean a decimal

0.b_1\cdots b_m\overline{a_1\cdots a_k}

which eventually repeats, but which cannot be reduced to a pure repeating decimal. An example is

.011363636\cdots = \tfrac{1}{88}

.
Prove that if a mixed repeating decimal is written as a fraction

\tfrac pq

in lowest terms, then the denominator

q

is divisible by

2

5

or both.

Problem Statement