Late last month, we have talked about fractions with terminating decimals as well fractions with non-terminating decimals. We ended up with a conjecture that a fraction is a terminating decimal if its denominator has only the following factors: 2 (or its powers), 5 (or its powers) or both. In this post,we refine this conjecture. This conjecture is the same as saying

A rational fraction in the lowest terms has a terminating decimal *if and only if* the integer has no prime factor other than and .

Note that we have already explained the *only if* part in the preceding post. It remains to show that if part which is

if is in lowest terms and contains at most and as factors, then the fraction is a terminating decimal.

For a specific example, we have

which is equal to

.

Now let us prove this statement.

Suppose the denominator where and are non-negative integers. There are two possible cases: or .

If , then we multiply the denominator by . This means that

.

This result can be simplified further to

.

Now, since is an integer and and are non-negative integers, the numerator is an integer and the denominator is a power of . Therefore, is a terminating decimal.

Now for the second case, let .

which is equal to

.

Again, is an integer, and are integers, and the denominator is a power of . Therefore, the fraction is a terminating decimal.

*Reference: Rational and Irrational Numbers by Ivan Niven*