Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
or
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
 
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Easy level:

Divisibility by Eleven

It is easy to tell that the following are multiples of 11: 22, 33, 44, 55, etc. But how about: 2728, or 31415? Are they divisible by 11?

Here an easy way to test for divisibility by 11. Take the alternating sum of the digits in the number, read from left to right. If that is divisible by 11, so is the original number.

So, for instance, 2728 has alternating sum of digits 2-7+2-8 = -11. Since -11 is divisible by 11, so is 2728.

Similarly, for 31415, the alternating sum of digits is 3-1+4-1+5 = 10. This is not divisible by 11, so neither is 31415.

Presentation Suggestions:
Students may enjoy thinking about how this divisibility test is related to the Fun Fact Divisibility by Seven.

The Math Behind the Fact:
This curious fact can be easily shown using modular arithmetic. Since 10n is congruent to (-1)n mod 11, we see that 1, 100, 10000, 1000000, etc. have remainders 1 when divided by 11, and 10, 1000, 10000, etc. have remainders (-1) when divided by 11. Thus

2728= 2*1000+7*100+2*10+8,

so its remainder when divided by 11 is just 2(-1)+7(1)+2(-1)+8(1), the alternating sum of the digits. (It's sum is the negative of what we found above because the alternation here begins with a -1.) But either way, if this alternating sum is divisible by 11, then so is the original number.

In fact, our observation shows more: that in fact when we take the alternating sum of the digits read from right to left (so that the sign of the units digit is always positive), then we obtain N mod 11.

How to Cite this Page:
Su, Francis E., et al. "Divisibility by Eleven." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Subjects:    number theory
Level:    Easy
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
4.22
 
current
rating
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!