# Obtain the check digits in this problem

• Math100
In summary, obtaining check digits is important for maintaining accurate and reliable data, especially when dealing with large amounts of information. Check digits are calculated using mathematical algorithms, such as the Luhn algorithm, and can be used for any type of numerical data. If there is an error in the check digit, it indicates an error in the data and prompts users to double-check for accuracy. There are different methods for obtaining check digits, each with their own set of rules and calculations.
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.

Math100

## 1. What is the purpose of obtaining check digits?

The purpose of obtaining check digits is to ensure the accuracy and validity of a numerical code or identification number. Check digits are used as a form of error detection, allowing for the identification of any errors or discrepancies in the code.

## 2. How are check digits calculated?

Check digits are calculated using a specific algorithm or formula, depending on the type of code or number being used. This algorithm typically involves performing mathematical operations on the digits of the code, resulting in a single check digit that can be used for verification purposes.

## 3. What types of codes or numbers require check digits?

Check digits are commonly used in identification numbers such as credit card numbers, barcodes, and ISBNs. They are also used in some government-issued identification numbers, such as social security numbers, to prevent fraud and ensure accuracy.

## 4. Can check digits be manually calculated?

Yes, check digits can be manually calculated using the appropriate algorithm or formula. However, this can be time-consuming and prone to errors. Many computer programs and online tools are available to automatically generate check digits for various types of codes and numbers.

## 5. What happens if a check digit does not match the rest of the code?

If a check digit does not match the rest of the code, it indicates that an error has occurred during data entry or transmission. This could be due to a mistake or intentional tampering. In this case, the code is considered invalid and further investigation is necessary to determine the cause of the error.

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