Subfield: Number Theory


Question: Call a rational number short if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m$-tastic if there exists a number $c \in\{1,2,3, \ldots, 2017\}$ such that $\frac{10^{t}-1}{c \cdot m}$ is short, and such that $\frac{10^{k}-1}{c \cdot m}$ is not short for any $1 \leqslant k<t$. Let $S(m)$ be the set of $m$-tastic numbers. Consider $S(m)$ for $m=1,2, \ldots$ What is the maximum number of elements in $S(m)$ ?

Solution:
First notice that $x \in \mathbb{Q}$ is short if and only if there are exponents $a, b \geqslant 0$ such that $2^{a} \cdot 5^{b} \cdot x \in \mathbb{Z}$. In fact, if $x$ is short, then $x=\frac{n}{10^{k}}$ for some $k$ and we can take $a=b=k$; on the other hand, if $2^{a} \cdot 5^{b} \cdot x=q \in \mathbb{Z}$ then $x=\frac{2^{b} \cdot 5^{a} q}{10^{a+b}}$, so $x$ is short.

If $m=2^{a} \cdot 5^{b} \cdot s$, with $\operatorname{gcd}(s, 10)=1$, then $\frac{10^{t}-1}{m}$ is short if and only if $s$ divides $10^{t}-1$. So we may (and will) suppose without loss of generality that $\operatorname{gcd}(m, 10)=1$. Define

$$
C=\{1 \leqslant c \leqslant 2017: \operatorname{gcd}(c, 10)=1\} \text {. }
$$

The $m$-tastic numbers are then precisely the smallest exponents $t>0$ such that $10^{t} \equiv 1$ $(\bmod \mathrm{cm})$ for some integer $c \in C$, that is, the set of orders of 10 modulo $\mathrm{cm}$. In other words,

$$
S(m)=\left\{\operatorname{ord}_{c m}(10): c \in C\right\}
$$

Since there are $4 \cdot 201+3=807$ numbers $c$ with $1 \leqslant c \leqslant 2017$ and $\operatorname{gcd}(c, 10)=1$, namely those such that $c \equiv 1,3,7,9(\bmod 10)$,

$$
|S(m)| \leqslant|C|=807
$$

Now we find $m$ such that $|S(m)|=807$. Let

$$
P=\{1<p \leqslant 2017: p \text { is prime, } p \neq 2,5\}
$$

and choose a positive integer $\alpha$ such that every $p \in P$ divides $10^{\alpha}-1$ (e.g. $\alpha=\varphi(T), T$ being the product of all primes in $P$ ), and let $m=10^{\alpha}-1$.

Claim. For every $c \in C$, we have

$$
\operatorname{ord}_{c m}(10)=c \alpha \text {. }
$$

As an immediate consequence, this implies $|S(m)|=|C|=807$, finishing the problem. Proof. Obviously $\operatorname{ord}_{m}(10)=\alpha$. Let $t=\operatorname{ord}_{c m}(10)$. Then

$$
c m\left|10^{t}-1 \Longrightarrow m\right| 10^{t}-1 \Longrightarrow \alpha \mid t \text {. }
$$

Hence $t=k \alpha$ for some $k \in \mathbb{Z}_{>0}$. We will show that $k=c$.

Denote by $\nu_{p}(n)$ the number of prime factors $p$ in $n$, that is, the maximum exponent $\beta$ for which $p^{\beta} \mid n$. For every $\ell \geqslant 1$ and $p \in P$, the Lifting the Exponent Lemma provides

$$
\nu_{p}\left(10^{\ell \alpha}-1\right)=\nu_{p}\left(\left(10^{\alpha}\right)^{\ell}-1\right)=\nu_{p}\left(10^{\alpha}-1\right)+\nu_{p}(\ell)=\nu_{p}(m)+\nu_{p}(\ell)
$$

so

$$
\begin{aligned}
c m \mid 10^{k \alpha}-1 & \Longleftrightarrow \forall p \in P ; \nu_{p}(\mathrm{~cm}) \leqslant \nu_{p}\left(10^{k \alpha}-1\right) \\
& \Longleftrightarrow \forall p \in P ; \nu_{p}(m)+\nu_{p}(c) \leqslant \nu_{p}(m)+\nu_{p}(k) \\
& \Longleftrightarrow \forall p \in P ; \nu_{p}(c) \leqslant \nu_{p}(k) \\
& \Longleftrightarrow c \mid k .
\end{aligned}
$$

The first such $k$ is $k=c$, so $\operatorname{ord}_{c m}(10)=c \alpha$.

Final answer: 807