- #1

Math100

- 771

- 219

- 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 ##.

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 ##.