Length of Reciprocal of Product of Powers of 2 and 5
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Theorem
Let $n \in \Z$ be an integer.
Let $\dfrac 1 n$, when expressed as a decimal expansion, terminate after $m$ digits.
Then $n$ is of the form $2^p 5^q$, where $m$ is the greater of $p$ and $q$.
Proof
Since $\dfrac 1 n$ terminates after $m$ digits:
From the first condition, we have $n = 2^p 5^q$ for some positive integers $p, q \le m$.
This gives $m \ge \max \set {p, q}$.
From the second condition, we cannot have both $p, q \le m - 1$.
Therefore at least one of $p, q$ is equal to $m$.
This gives $m \le \max \set {p, q}$.
These results give $m = \max \set {p, q}$.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$