Obtain the check digits in this problem

[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
$a_{9}\equiv (7a_{1}+3a_{2}+9a_{3}+7a_{4}+3a_{5}+9a_{6}+7a_{7}+3a_{8})\pmod {10}$.
Obtain the check digits that should be appended to the two numbers $55382006$ and $81372439$.

Relevant Equations

None.

First, we consider the integer $55382006$.
Observe that
\begin{align*}
&a_{9}\equiv (5\cdot 7+5\cdot 3+3\cdot 9+8\cdot 7+2\cdot 3+0\cdot 9+0\cdot 7+6\cdot 3)\pmod {10}\\
&\equiv 157\pmod {10}\\
&\equiv 7\pmod {10}.\\
\end{align*}
Thus $a_{9}=7$.
Next, we consider the integer $81372439$.
Observe that
\begin{align*}
&a_{9}\equiv (8\cdot 7+1\cdot 3+3\cdot 9+7\cdot 7+2\cdot 3+4\cdot 9+3\cdot 7+9\cdot 3)\pmod {10}\\
&\equiv 225\pmod {10}\\
&\equiv 5\pmod {10}.\\
\end{align*}
Thus $a_{9}=5$.
Therefore, the check digits that should be appended to the two numbers $55382006$ and $81372439$ are $7$ and $5$.

 

[fresh_42]

Correct.