Charles L. Dodgson, better known as Lewis Carroll, gave the following method in 1897 for doing just that.
Given a number to test, delete its units digit and then subtract this digit from the shortened number. For example, if you were to test 31467898, the new number you get would be 3146789 - 8 = 3146781. Since the sum of the two numbers is a multiple of 11 (you can check this fact), the new number is divisible by 11 if and only if the original number is. Now repeat the procedure till you get a number that you know is or is not divisible by 11. For example, with our example, the next numbers you get would be:
|314677||=||314678 - 1|
|31460||=||31467 - 7|
|3146||=||3146 - 0|
|308||=||314 - 6|
|22||=||30 - 8|
There are other ways too: If our number is anan-1¼a2a1a0 then it is divisible by 11 if and only if a0-a1+a2-¼+(-1)nan is also divisible by 11. Applying this test to the same example, we find that 31467898 is divisible by 11 if and only if 3-1+4-6+7-8+9-8=0 is divisible by 11, which of course it is.
Let's move to something more familiar - checking if a number is divisible by 9. The method commonly taught in schools is to add all its digits - their sum is a multiple of 9 if and only if the original number is. So for instance 1234565 is not a multiple of 9 since 1+2+3+4+5+6+5=26 is not a multiple of 9.
The question is, why do these methods work? Why is the test for 9 much easier to use than that for 11? What's so special about these numbers? Can we always find tests (even difficult ones) for any divisibility by any number, say 13 or 17? I confess I do not know the answer to the last question. But the first step to begin answering such questions is to run away from decimals. Why should we always represent numbers in base 10? What happens in other bases? Consider say, base 7. Can you devise methods that tell you whether a number represented in this base is divisible by 6? by 810 = 117? by 910 = 127? More next time, if the Editor reminds me.
There is plenty of Lewis Carroll stuff on the web. Plenty of links can be found here. More information on the work of Lewis Carroll can be found at the excellent site of the University of Waterloo Computer Science Club. Since this year is the hundredth anniversary of his death, we give a bit of info on him.
He was born in 1833 to a clergyman. His received a good education, culminating in an excellent degree at Oxford in 1854. He stayed there as a lecturer till 1881, during which time he wrote several student texts. He will be better remembered however for the books ``Alice's adventures in Wonderland'' (1865) and ``Through the looking glass'' (1872).
The two books were originally written for Alice Liddell, a child of a friend who loved listening to his stories, despite his stammer. Charles was a very good amateur photographer and Alice appears in several of his pictures. The books were well received by the general public. Even Queen Victoria requesting a copy of the next book by the same writer. Unfortunately for her, this turned out to be the Syllabus of Plane Algebraical Geometry.!
Picture ripped off from the St Andrew's site.