Casting Out Elevens - Trachtenberg Speed Math

The elevens remainder or "casting out elevens" as it is also known can be used as a double check on the casting out nines or simply as a different way to check your results.

In the elevens remainder we do not actually divide by 11 and the method used will depend on the number of digits in the number.

Case 1: Two Digit Numbers

For two digit numbers the rules are:
Subtract the tens digit from the unit digit. Tens and Units digits If the unit digit is smaller than the tens digit then add 11 to the units digit before subtracting the tens digit.
Lets look at some examples to make this clearer.

$\textcolor{red}{5}\textcolor{blue}{7}$

We subtract $\textcolor{red}{5}$ from $\textcolor{blue}{7}$ which gives us $\boldsymbol{2}$ .
(note: 57 divided by 11 is 5 with 2 remainder)

$\textcolor{red}{9}\textcolor{blue}{3}$

The $\textcolor{blue}{3}$ is less than $\textcolor{red}{9}$ so we first add $11$ to $\textcolor{blue}{3}$ to get 14, now we subtract $\textcolor{red}{9}$ from $14$ which is $\boldsymbol{5}$ .
(note: 93 divided by 11 is 8 with 5 remainder)

Case 2: All Numbers Longer Than Two Digits

For these numbers we start at the right hand side and we sum up the last digit and every second digit moving to the left and come up with our first total. Then we go back to the right hand side and sum up the second last digit and very second digit moving to the left to give us our second total. We then subtract the second total from the first.

Lets look at an example:

$537,682,910$

We will color code the digits to help with the explanation

$\textcolor{blue}{5}\textcolor{red}{3}\textcolor{blue}{7},\textcolor{red}{6}\textcolor{blue}{8}\textcolor{red}{2},\textcolor{blue}{9}\textcolor{red}{1}\textcolor{blue}{0}$

From the right we will add the digits in blue: $\textcolor{blue}{0} + \textcolor{blue}{9} + \textcolor{blue}{8} + \textcolor{blue}{7} + \textcolor{blue}{5} = \textcolor{blue}{29}$

Now from the right we add up the red digits: $\textcolor{red}{1} + \textcolor{red}{2} + \textcolor{red}{6} + \textcolor{red}{3} = \textcolor{red}{12}$

We now subtract the second total (red digits) from the first total (blue digits): $\textcolor{blue}{29} - \textcolor{red}{12} = 17$

The $17$ still needs to be reduced, as it is larger than $11$ , so we follow the two digit method and subtract the tens digit from the unit digit.

$7 - 1 = 6$

So $6$ is the elevens remainder of $537,682,910$ .

How you go about adding and subtracting the numbers to find the elevens remainder does not have to only follow the two pass method above.

Variation 1: Add first pass then subtract each alternate digit

One alternative is after doing the first addition above, of the blue digits, when we come to doing the red digits we could have used our blue digit total and subtracted each red digit from that.

$\begin{equation*} \begin{split} \textcolor{Blue}{29} - \textcolor{red}{1} &= 28\\ 28 - \textcolor{red}{2} &= 26\\ 26 - \textcolor{red}{6} &= 20\\ 20 - \textcolor{red}{3} &= 17\\ 7 - 1 &= 6 \end{split} \end{equation*}$

Variation 2: Treat as several two digit pairs then add the results

Another effective shortcut with large numbers is to split the number into digit pairs.
If we take the number 46,389,205 and split it into digit pairs as seen below where we underline each pair. Eleven remainder case 2 digit pairs We now treat each pair as a two digit number and subtract the tens figure from the units figure, or we subtract the left hand side digit from the right hand side digit.

$\begin{equation*} \begin{split} \text{the number pair $46$ gives } 6 - 4 &= 2\\ \text{the number pair $38$ gives } 8 - 3 &= 5\\ \text{the number pair $92$ gives } 2 + 11 &= 13\\ 13 - 9 &= 4\\ \text{the number pair $05$ gives } 5 - 0 &= 5 \end{split} \end{equation*}$

Now we add up the results from each pair

$2 + 5 + 4 + 5 = 16$

We need to reduce the 16, as it is larger than 11, and we do this by subtracting the tens digit from the units digit.

$6 - 1 = 5$

Variation 3: Alternately add and subtract as you go

Another way to calculate the elevens remainder is to start at the right hand digit and alternate between subtracting and adding the digits so we make only one pass.Looking at our previous example we would add blue digits and subtract red digits.

$\textcolor{blue}{5}\textcolor{red}{3}\textcolor{blue}{7},\textcolor{red}{6}\textcolor{blue}{8}\textcolor{red}{2},\textcolor{blue}{9}\textcolor{red}{1}\textcolor{blue}{0}$

So we would have $\textcolor{blue}{0} + 11 - \textcolor{red}{1} + \textcolor{blue}{9} - \textcolor{red}{2} + \textcolor{blue}{8} - \textcolor{red}{6} + \textcolor{blue}{7} - \textcolor{red}{3} + \textcolor{blue}{5} = 28$

Then $8 - 2 = 6$

Note that since we cant subtract $1$ from $0$ , we add $11$ to the $0$ first.

If you find your total getting above $20$ you can subtract $11$ from it to keep the addition or subtraction simple. This will not affect the outcome of the digit root.
Similar to the casting out nines we apply the casting out elevens as a check on calculations.

Casting out Elevens In Multiplication

Lets look at $243 \times 257$

$243 \times 257 = 62451$

We use the Case 2 method to find each elevens remainder:

$\begin{equation*} \begin{split} 62451&:\quad1 + 4 + 6 - 5 - 2 = 4\\ 243&:\quad2 + 3 - 4 = 1\\ 257&:\quad7 + 2 - 5 = 4 \end{split} \end{equation*}$

$1 \times 4 = 4$

So our check is okay.
Multiplying the elevens remainders for the multiplicand and the multiplier gives us a result of $4$ which is also the elevens remainder of the product so our calculation is correct.

Casting Out Elevens In Addition

We will do a simple sum.

$23 + 54 = 77$

We will use the case 1 method for 2 digit numbers:

$\begin{equation*} \begin{split} 77&:\quad7 - 7 = 0\\ 23&:\quad3 - 2 = 1\\ 54&:\quad4 + 11 - 5 = 10 \end{split} \end{equation*}$

Adding the elevens remainders for 23 and 54 we get:

$\begin{equation*} \begin{split} 1 + 10 = 11\\ 1 - 1 = 0 \end{split} \end{equation*}$

(As the remainder sum is greater than ten we use the rule for two digit numbers and subtract the tens digit from the unit digit, in this case $1 - 1$ )
So we get $0$ on both sides so our calculation is correct

Casting Out Elevens In Division

To check division we use the following procedure:

If there is a remainder, subtract it from the dividend first to get the reduced dividend.
Find the elevens remainder of the dividend or reduced dividend.
Find the elevens remainder of the divisor.
Find the elevens remainder of the answer.
Multiply the elevens remainders of the dividend and the answer.
Compare the product of the step 5 with result of step 2.

Some Examples.

$\text{A) }4272 \div 24 = 178 \quad\text{with no remainder}$

There is no remainder, skip to 2.
The elevens remainder of the dividend:
$\textcolor{red}{4}\textcolor{blue}{2}\textcolor{red}{7}\textcolor{blue}{2} = (\textcolor{blue}{2} + \textcolor{blue}{2}) + 11 - (\textcolor{red}{7} + \textcolor{red}{4}) = 4$
The elevens remainder of the divisor:
$24 = 4 - 2 = 2$
The elevens remainder of the answer:
$178 = (8 + 1) - 7 = 2$
Multiply the elevens remainders of the divisor and the answer:
$2 \times 2 = 4$
The elevens remainder in steps 2 and 5 are the same so our result is correct.

$\text{B) }23534 \div 35 = 672 \quad\text{with 14 remainder}$

Subtract the remainder from the dividend:
$23534 - 14 = 23520$
The elevens remainder of the reduced dividend:
$\textcolor{blue}{2}\textcolor{red}{3}\textcolor{blue}{5}\textcolor{red}{2}\textcolor{blue}{0} = (\textcolor{blue}{0} + \textcolor{blue}{5} + \textcolor{blue}{2}) - (\textcolor{red}{2} + \textcolor{red}{3}) = 2$
The elevens remainder of the divisor:
$35 = 5 - 3 = 2$
The elevens remainder of the answer:
$672 = 2 + 6 - 7 = 1$
Multiplying the elevens remainders of the divisor and the answer:
$2 \times 1 = 2$
The elevens remainder in steps 2 and 5 are the same so our result is correct.

Casting Out Elevens In Subtraction

Let's look at $321 - 54 = 267$
The elevens remainders are:

$\begin{equation*} \begin{split} 321 &= (1 + 3 - 2) = \textcolor{blue}{2}\\ 54 &= (4 + 11 - 5) = \textcolor{red}{10}\\ 267 &= (7 + 2 - 6) = 3 \end{split} \end{equation*}$

$\textcolor{blue}{2} + 11 - \textcolor{red}{10} = 3$

We subtract the elevens remainder of 54 from the elevens remainder of 321, however 2 is less than 10 so we add 11 to the elevens remainder of 321 giving us 13. then we subtract the 10 from the elevens remainder of 54 giving us 3 the same value as the elevens remainder of 267 so our calculation is correct.

With subtraction you also have the choice to turn it into an addition, using the above example:

$321 - 54 = 267$

We can turn this into an addition and using the same remainders we got above

$267 + 54 &= 321$

$\begin{equation*} \begin{split} 3 + 10 &= 13\\ 3 - 1 &= 2\quad\text{elevens remainder of }13 \end{split} \end{equation*}$

Adding the elevens remainders for $267$ and $54$ we get $13$ and the elevens remainder of that is $2$ the same as the elevens remainder for $321$ so it is correct.

Practice Casting Out Elevens

I know when explaining how to do this it looks like a lot of work but once you understand the concept and with a little practice you can quickly do some quality checking on your results. So before we go how about testing what you have learnt and try to answer the questions below.

Q1 : What is the elevens remainder of the following numbers?

$\begin{equation*} \begin{split} &\text{a) } 7428\\ &\text{b) } 3815\\ &\text{c) } 3819351 \end{split} \end{equation*}$

Q2: Are the following equations correct? Only check the results by casting out elevens.

$\begin{align*} &\text{a) } 7463 + 9134 = 16597\\ &\text{b) } 234 \times 537 = 126558\\ &\text{c) } 73521 \div 27 = 2723 \end{align*}$

No peeking! Try the questions before you scroll down further to see the answers.

Answers:
A1:

$\begin{align*} 7428 &= 8 - 2 + 4 - 7 = 3\\ 3815 &= 5 + 8 - 1 - 3 = 9\\ 3819351 &= 1 + 3 + 1 + 3 - 5 + \textcolor{blue}{11} - 9 + \textcolor{blue}{11} - 8 = 8 \end{align*}$

note we had to add 11 twice here to be able to subtract 9 and then 8

A2:

$\begin{align*} \text{a) } 7463 + 9134 &= 16597\\ 5\quad +\enskip 4\quad &=\quad 9\\ 7463 &= 3 + 11 - 6 + 4 - 7 = 5\\ 9134 &= 4 - 3 + 1 + 11 - 9 = 4\\ 16597 &= 7 + 11 - 9 + 5 - 6 + 1 = 9 \end{align*}$

$\begin{align*} \text{b) } 234 \times 537 &= \textcolor{red}{126558}\\ \quad3\enskip\times\enskip 9\enskip &\textcolor{white}{=}\quad\enskip 3\\ 27\quad\quad &\textcolor{white}{=} \\ 5\quad\quad &\ne \quad\enski 3 \quad\text{The correct answer is \textcolor{blue}{125658}} \end{align*}$

$\begin{align*} 234 &= 4 - 3 + 2 = 3\\ 537 &= 7 - 3 + 5 = 9\\ \textcolor{red}{126558} &= 8 - 5 + 5 - 6 + 2 - 1 = 3\\ 9 &\times 3 = 27\\ 27 &= 7 - 2 = 5\\ \textcolor{blue}{125658} &= 8 - 5 + 6 - 5 + 2 - 1 = 5 \end{align*}$

$\begin{align*} \text{c) } 73521 \div 27 &= 2723\qquad \text{(rearrange into a multiplication)}\\ 2723 \times 27 &= 73521\\ 6\enskip \times\enskip 5 & \qquad 8\\ 30\quad&\\ 30 = 0 &+ 11 - 3 = 8 \end{align*}$