You all knew that an even integer is divisible by two, and you probably all knew the rule for divisibility by three: that if the sum of the digits is divisible by three, then so is the number itself. Do you know the rule for divisibility by 9, or by 11?
These are curious rules, because they're based in part on the base of the number system in use. As we saw in the case of the sieve of Eratosthenes, the spiraling patterns that developed were a consequence of the wrapping of the integers at every 10th.
This section delves into bases, and proves that these rules for divisibility work.
base b place-value notation: the representation of a number as as string of coefficients of powers of b (e.g. decimal notation, or binary - base 2 - number system):
which stands for
where 
and 
.
You've got to hear Tom Lehrer do a subtraction problem (342 - 173) in his song ``New Math''.
Theorem: Given any integer b>1, any integer may be written uniquely in base b place-value notation.
Proof: repeated applications of the division algorithm.
Theorem 4.4: Let 
be a polynomial function of
x with integral coefficients 
.
If 
, then 
.
Proof:
Therefore,
and the sums of all the coefficients are equal as well, i.e. .
Corollary: If a is a solution of the congruence
, and , then b is also a
solution.
Theorem 4.5/4.6: Let
be the decimal expansion of positive integer N, 
, and let
. Then
-
.
-
Let
. Then
.
Often ``the trick'' to solving the problems involves
- figuring out which n in ``modulo n'' we need, or
- figuring out how to rewrite things modulo n so that good things happen.