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I need help determining what operation I would use to determine the check digit of this equation \begin{eqnarray*} x_{11} = x_1 + x_2 + x_3 + \cdots + x_{10} \pmod{9} \end{eqnarray*}

Those are sub numbers, rather than $x \times 11$. I don't know how to format them correctly. I'm a novice

Donald Splutterwit
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Johnny James
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  • I don't really understand your question. Could you give us an example ? – Donald Splutterwit Oct 11 '17 at 20:51
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    I agree with @Donald, which is your actual question? – cronos2 Oct 11 '17 at 20:53
  • I'm supposed to determine what x11 is equal to – Johnny James Oct 11 '17 at 20:57
  • It says. "The first ten digits (x1 through x10)are an identifier; the final digit is a check digit that satisfies the following", then it lists the equation. It's a textbook exercise and it's confusing me. The answers listed to choose from, are 1, 4, 5 or 8. I haven't a clue what operation I should be using. It gives the numbers "7555618873_" I'm assuming the blank is where x11 goes – Johnny James Oct 11 '17 at 21:00
  • If you assume all the ten digits are distinct and non-zero then $x_{11}=1\pmod9$. If not, you have a lot of possible solutions; in particular if all the ten digits are equal you have $10x=9x+x= x\pmod 9$ where $x\in{1,2,3,4,5,6,7,8,9}$. – Piquito Oct 11 '17 at 21:21

2 Answers2

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In the comments you wrote:

It says. "The first ten digits (x1 through x10)are an identifier; the final digit is a check digit that satisfies the following", then it lists the equation. It's a textbook exercise and it's confusing me. The answers listed to choose from, are 1, 4, 5 or 8. I haven't a clue what operation I should be using. It gives the numbers "7555618873_" I'm assuming the blank is where x11 goes

So in this $x_1=7; x_2 =5; x_3 = 5;..... x_{10}=3$

So $x_{11} = 7+5+5+5+6+1+8+8+7+3 \mod 9$.

"casting out nines" I get $7\equiv -2;5+5\equiv 1;5+6\equiv 2; 1+8\equiv 0;8\equiv -1;7+3 \equiv 1$ so $-2+1+2-1+1 \equiv 1\mod 9$

So $x_{11} = 1$

So the number is $75556188731$

The little "subnumber" $i$ in $x_i$ is just a way of saying "we are going to refer to the $i$th digit as $x_i$". Instead of writing:

Let $a$ be the first digit. Let $b$ be the second digit. .... Let $\overline{\not{\mu}}$ be the 478th digit. Let....

Simply say:

Let $\{x_i\}$ be a set of variables so that $x_i$ represents that $i$-th digit.

The actual values of the $i$ (called the "index") doesn't have anything to do with the value or any operation. It is merely a way of saying, it's the $i$th item in a list.

fleablood
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Hint: Simply do what the defining equation says: add the numbers $$7+5+5+5+6+1+8+8+7+3$$ and take as $x_{11}$ the remainder after dividing the sum by $9$.

gammatester
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  • That was my first thought, but the sub numbers threw me off. Do the sub numbers not factor into the equation at all? – Johnny James Oct 11 '17 at 21:32
  • What is a "sub number"? – fleablood Oct 11 '17 at 21:34
  • X sub 1 would be X1 or rather a small 1 beside the x. I don't know how to put it here – Johnny James Oct 11 '17 at 21:39
  • Normally the 'subnumbers' are called digits and you compute the digit sum. And $x_{11}$ is the check digit (as already named in the task). You enter them as e.g. $x_{11}$. – gammatester Oct 11 '17 at 21:39
  • Those are just indexes. They help you tell which number is which. Replace them with $a,b,c,d,e,f,g,h,i,j$ if you wish. But this way you know $x_7$ is the seventh digit. But the indexes themselves don't have anything to do with anything. – fleablood Oct 11 '17 at 21:47
  • @fleablood: I thought, the $x_i$ are the sub-numbers (part of the 10-digit number), and not the indeces $i,$ but your interpretation makes sense. – gammatester Oct 11 '17 at 22:13