Clay Kitchings ::
EMAT 6600 :: Divisibility

Again, this
problem arose from curiosity about rules of divisibility for various integers
in MATH 6000. Where do rules for
divisibility come from? How did
ÒtheyÓ come up with them? (And who on earth is/are ÒtheyÓ anyway?)

Rules for
divisibility come from ÒexpandingÓ numbers and writing them almost as if they
were polynomials. For example, one can think of 2532 as the following: .

** Problem**: Generate the rule for divisibility
by 11.

Some
experience with modular arithmetic comes in handy for this derivation. We shall
apply some properties of modular arithmetic and see what happens:

Now, apply
the definition of congruence in modular arithmetic:

But, what
does this summation really mean? It means:

Basically,
it is a +/- alternating ÒsumÓ of the digits. So, 11|n if and only if 11 divides the
Òalternating sumÓ of the digits.
Here is an example:

Is
3417524 divisible by 11?

Check: 4-2+5-7+1-4+3 = 0, which is certainly
divisible by 11. Therefore 3417524 is divisible by
11.