The Dean's Column
The Arithmetic Uniqueness of the Number 11
By Alfred S. Posamentier, Ph.D.
The number 11, since it is 1 greater than our base 10, has some lovely properties that can be used not only to shortcut some calculations, but also to exhibit some of mathematics’ hidden treasures. We begin with considering a method to multiply by 11 mentally and then inspect when a number is divisible by 11.
This is a very nifty way to multiply by 11. This one always gets a rise out of students, because it is so simple – and, believe it or not, even easier than doing it on a calculator!
The rule is very simple: To multiply a two-digit number by 11 just add the two digits and place the sum between the two digits.
For example, suppose you need to multiply 45 by 11. According to the rule, add 4 and 5 and place it between the 4 and 5 to get 495. It’s as simple as that.
This can get a bit more difficult, as students will be quick to point out. If the sum of the two digits is greater that 9, then we place the units digit between the two digits of the number being multiplied by 11 and “carry” the tens digit to be added to the hundreds digit of the multiplicand. Let’s try it with 78·11
7+8=15. We place the 5 between the 7 and 8, and add the 1 to the 7, to get [7+1][5][8] or 858.
Your students will next request that you extend this procedure to numbers of more than two digits.
Let’s go right for a larger number such as 12,345 and multiply it by 11.
Here we begin at the right side digit and add every pair of digits going to the left.
1[1+2][2+3][3+4][4+5]5 = 135,795.
If the sum of two digits is greater than 9, then use the procedure described before: place the units digit appropriately and carry the tens digit. We will do one of these for you here. Multiply 456,789 by 11.
We carry the process step by step:
4[4+5][5+6][6+7][7+8][8+9]9
4[4+5][5+6][6+7][7+8][17]9
4[4+5][5+6][6+7][7+8+1][7]9
4[4+5][5+6][6+7][16][7]9
4[4+5][5+6][6+7+1][6][7]9
4[4+5][5+6][14][6][7]9
4[4+5][5+6+1][4][6][7]9
4[4+5][12][4][6][7]9
4[4+5+1][2][4][6][7]9
4[10][2][4][6][7]9
[4+1][0][2][4][6][7]9
[5][0][2][4][6][7]9
5,024,679
Students will be enthusiastic with this procedure, because it is so simple. They will go home and show it to their family and friends. By showing it and doing it, it will stay with them. Your goal is to maintain this enthusiasm.
Now having convinced students that the number 11 has a special property, have them consider when a number is divisible by 11 – that is determining this without actually doing the division. Try to convince students that at the oddest times this question can come up. If you have a calculator at hand, the problem is easily solved. But that is not always the case. Besides, there is such a clever “rule” for testing for divisibility by 11 that it is worth showing students just for its charm.
The rule simply states:
If the difference of the sums of the alternate digits is divisible by 11, then the original number is also divisible by 11.
Sounds a bit complicated, but it really isn’t. Have your students take this rule a piece at a time. The sums of the alternate digits means you begin at one end of the number taking the first, third, fifth, etc. digits and add them. Then add the remaining (even placed) digits. Subtract the two sums and inspect for divisibility by 11.
It is probably best shown to your students by example. We shall test 768,614 for divisibility by 11.*
Sums of the alternate digits are: 7+8+1 = 16, and 6+6+4 = 16. The difference of these two sums, 16-16 = 0, which is divisible by 11.
Another example might be helpful to firm up your student’s understanding. To determine if 918,082 is divisible by 11, find the sums of the alternate digits:
9+8+8 = 25, and 1+0+2 = 3.
Their difference is 25-3 = 22, which is divisible by 11, and so the number 918,082 is divisible by 11.
Now just let your students practice with this rule. Once again, practice with this procedure will insure its permanence with students, who throughout their mathematics study should be building an arsenal of tools with which to navigate through their further study in mathematics. Above all, with these little tidbits they will allow themselves to be charmed by the subject that all too often does not enjoy appropriate popularity in our society.
Dr. Alfred Posamentier is Dean of the School of Education at City College of NY, author of over 40 Mathematics books, including: Math Wonders to Inspire Teachers and Students (ASCD, 2003) and The Fabulous Fibonacci Numbers (Prometheus, 2007), and member of the NYS Mathematics Standards Committee.
*Remember (0/11) = 0
