Finding the prefix length of a decimal expansion

Remember from my previous post that we’re trying to find the prefix length $k$ and repetend length $n$ of the decimal expansion of a fraction $a/b$ , that is, the length of the part before it starts repeating, and the length of the repeating part. In that post I showed how to reduce it to the following question:

What are the smallest values of $k$ and $n$ such that $10^k \cdot (10^n - 1)$ is evenly divisible by $b$ ?

and I left it at that, with some questions for thought.

Can you see why we get to determine the values of $k$ and $n$ separately, i.e. neither value influences the other at all?

The reason we get to consider $k$ and $n$ separately is that $10^k$ and $(10^n - 1)$ cannot possibly share any prime factors in common. $10^k = 2^k 5^k$ only has 2 and 5 as prime factors; on the other hand, $(10^n - 1)$ cannot have 2 or 5 as prime factors since it is equivalent to $(-1) \bmod 2$ and $\bmod 5$ . So the original question splits into two independent questions: (1) What is the smallest value of $k$ such that $10^k$ cancels all the factors of $2$ and $5$ in $b$ ? (2) What is the smallest value of $n$ such that $(10^n - 1)$ cancels all the other prime factors in $b$ ?

The first sub-question is easy enough to answer: if $b = 2^x 5^y b'$ , where $b'$ has no factors of $2$ or $5$ , then choosing $k = \max(x, y)$ is both necessary and sufficient: it will be just enough to cancel all the factors of $2$ and $5$ in $b$ .

Let’s see how this works. The example we looked at last time was $\displaystyle 89/420 = 0.21\overline{190476}$ . If we factor $420$ we get $(2^2 \cdot 5) \cdot (3 \cdot 7)$ , so our analysis above says $k = 2$ , since there are two factors of $2$ . And sure enough, the decimal expansion has a prefix of length $2$ .

As another example, let’s pick a denominator by its factorization: suppose we have a denominator of $2^2 \cdot 5^4 \cdot 11 \cdot 19 \cdot 37 = 19332500$ . 5 has a higher exponent than the 2, so we predict the prefix will have length 4: and sure enough, for example,