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,

\displaystyle 1/19332500 = 0.0000\overline{000517263675158412}.

We can try other numerators too, we just have to make sure they are relatively prime to 19332500. For example,

\displaystyle 1500007 / 19332500 = 0.0775\overline{899133583344109659}.

In my next post I’ll talk about how to find the repetend length!

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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3 Responses to Finding the prefix length of a decimal expansion

  1. Pavel says:

    In the second to last sentence, I think you meant trying other *numerators*, not *denominators*.

  2. Pingback: Finding the repetend length of a decimal expansion | The Math Less Traveled

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