Determine the value of the obscured digit

[Math100]

Homework Statement

Consider the eight-digit bank identification number $a_{1}a_{2}...a_{8}$, which is followed by a ninth check digit $a_{9}$ chosen to satisfy the congruence
$a_{9}\equiv (7a_{1}+3a_{2}+9a_{3}+7a_{4}+3a_{5}+9a_{6}+7a_{7}+3a_{8})\pmod {10}$.
The bank identification number $237a_{4}18538$ has an illegible fourth digit. Determine the value of the obscured digit.

Relevant Equations

None.

Consider the bank identification number $237a_{4}18538$.
Note that $a_{9}=8$.
This means
\begin{align*}
&a_{9}\equiv (2\cdot 7+3\cdot 3+7\cdot 9+a_{4}\cdot 7+1\cdot 3+8\cdot 9+5\cdot 7+3\cdot 3)\pmod {10}\\
&\equiv (205+7a_{4})\pmod {10}\\
&\equiv (5+7a_{4})\pmod {10}.\\
\end{align*}
Since $3-7a_{4}=10k$ for some $k\in\mathbb{Z}$ where $0\leq a_{4}\leq 9$,
it follows that $-63\leq -7a_{4}\leq 0\implies -60\leq 3-7a_{4}\leq 3$.
Thus $a_{4}=9$.
Therefore, the value of the obscured digit is $9$.

 

[fresh_42]

Homework Statement:: Consider the eight-digit bank identification number $a_{1}a_{2}...a_{8}$, which is followed by a ninth check digit $a_{9}$ chosen to satisfy the congruence
$a_{9}\equiv (7a_{1}+3a_{2}+9a_{3}+7a_{4}+3a_{5}+9a_{6}+7a_{7}+3a_{8})\pmod {10}$.
The bank identification number $237a_{4}18538$ has an illegible fourth digit. Determine the value of the obscured digit.
Relevant Equations:: None.

Consider the bank identification number $237a_{4}18538$.
Note that $a_{9}=8$.
This means
\begin{align*}
&a_{9}\equiv (2\cdot 7+3\cdot 3+7\cdot 9+a_{4}\cdot 7+1\cdot 3+8\cdot 9+5\cdot 7+3\cdot 3)\pmod {10}\\
&\equiv (205+7a_{4})\pmod {10}\\
&\equiv (5+7a_{4})\pmod {10}.\\
\end{align*}
Since $3-7a_{4}=10k$ for some $k\in\mathbb{Z}$ where $0\leq a_{4}\leq 9$,
it follows that $-63\leq -7a_{4}\leq 0\implies -60\leq 3-7a_{4}\leq 3$.
Thus $a_{4}=9$.
Therefore, the value of the obscured digit is $9$.

Right.