I'm currently carrying on a project based on the article by Frey and Hammet in 2021 College Maths Journal.
This article is written about how to generate divisibility tests from truncating the very final digit of a dividend, and there is a section regarding truncation of larger blocks of final digits which the authors did not give in the paper.
Let $\beta \geq 1$ be the amount of final digits which are going to be truncated using the "Last digit truncation method."
For any $m \in \mathbb{N}$ which $\gcd(m,10) = 1$, $$mx = 10y \pm c \iff x = \pm j + 10t, y = \pm k + mt, t \in \mathbb{Z}$$ and $$m \mid n \iff m \mid \frac{n - a_0}{10} \mp ya_0$$ where $mx = 10y \pm 1$ and $a_0$ is the last digit of $n$.
This is the algorithm in the case of $\beta = 1$.
Now, I have supposed the claim (Induction hypothesis): $$mx = 10^{\kappa}y \pm c \iff x = \pm j + 10^{\kappa}t, y = \pm k + mt, t \in \mathbb{Z}$$ and $$m \mid n \iff m \mid \frac{n - \sum^{\kappa - 1}_{i=0} 10^{i}a_i}{10^{\kappa}} \mp y\sum^{\kappa - 1}_{i=0} 10^{i}a_i $$ where $mx = 10^{\kappa}y \pm 1$ and $a_i$ is the last $i$ digits of $n$, is true for all $\beta = \kappa \geq 1$
It's easy to see that the first equivalence is true for $\kappa + 1$, or analogously, for all $\beta \geq 1$.
The problem is at the second equivalence.
Consider $n = a_{s}10^s + a_{s-1}10^{s-1} + ... + a_{1}10 + a_0$. From the claim, we know that $$m \mid n \iff m \mid \frac{n - \sum^{\kappa - 1}_{i=0} 10^{i}a_i}{10^{\kappa}} \mp y\sum^{\kappa - 1}_{i=0} 10^{i}a_i$$ We can then suppose that $\begin{equation*} \begin{split} n' & = \frac{n - \sum^{\kappa - 1}_{i=0} 10^{i}a_i}{10^{\kappa}} \mp y\sum^{\kappa - 1}_{i=0} 10^{i}a_i\\ & = a_{s}10^{s-\kappa} + a_{s-1}10^{s-\kappa-1} + ... + a_{\kappa+1}10 + a_{\kappa}\\ & \hspace{0.5cm} \mp y(a_{\kappa-1}10^{\kappa-1} + a_{\kappa-2}10^{\kappa-2} + ... +a_{1}10 + a_0) \end{split} \end{equation*}$
which implies $$m \mid n \iff m \mid n'$$ Without loss of generality, $\begin{equation*} \begin{split} m \mid n' & \iff m \mid \frac{n'-a_{\kappa}}{10} \mp ya_{\kappa}\\ & \iff m \mid \frac{n'-a_{\kappa}}{10} \mp ya_{\kappa}\\ & \iff m \mid \frac{(\frac{n - \sum^{\kappa-1}_{i=0} 10^{i}a_i}{10^{\kappa}}\mp y\sum^{\kappa - 1}_{i=0} 10^{i}a_i) - a_{\kappa}}{10}\mp ya_{\kappa}\\ & \iff m \mid \frac{n - \sum^{\kappa-1}_{i=0} 10^{i}a_i \mp 10^{\kappa}y\sum^{\kappa - 1}_{i=0} 10^{i}a_i - 10^{\kappa} a_{\kappa}}{10^{\kappa+1}}\mp ya_{\kappa}\\ & \iff m \mid \frac{n - \sum^{\kappa-1}_{i=0} 10^{i}a_i \mp 10^{\kappa}y\sum^{\kappa - 1}_{i=0} 10^{i}a_i - 10^{\kappa} a_{\kappa} \mp 10^{\kappa + 1}ya_{\kappa}}{10^{\kappa+1}}\\ & \iff m \mid \frac{n - \sum^{\kappa}_{i=0} 10^{i}a_i}{10^{\kappa+1}} \mp y\sum^{\kappa}_{i=0} 10^{i}a_i\\ \end{split} \end{equation*}$
Does my progress look strange? I feel like I have mistaken in the proof. Can somebody spot it for me? Send help!