Arithmetic – Detecting a Repeating Decimal in a Fraction

Question

Given any fraction where both the numerator (N) and denominator (D) are both positive and are both whole numbers.

Without manually dividing N by D, is it possible to pre-determine if the resulting value represented in decimal would be a repeating value? (e.g. 44÷33 is 1.3333333333….)

I believe the value of N ÷ D will NOT be a repeating decimal if and only if D is any of the following

1. D is equal to 1
   OR
2. D's prime factors only consist of 2's and/or 5's. (includes all multiples of 10)

Otherwise, if none of the two rules above hold true, then the positive whole numbers N and D will divide into repeating decimal.

Correct, or am I missing a case?

Answer 1

Correct.

From wikipedia:

A decimal representation written with a repeating final 0 is said to terminate before these zeros. Instead of "1.585000…" one simply writes "1.585". The decimal is also called a terminating decimal. Terminating decimals represent rational numbers of the form $k/(2^n5^m)$.

http://en.wikipedia.org/wiki/Repeating_decimals