We say one integer divides another if it does so evenly, that is
with a remainder of zero (we sometimes say, "with no remainder,"
but that is not technically correct). More formally,
mathematicians write: If a and b are integers (with a not zero),
we say a divides b if there is an integer c such that b = ac. We
use this concept enough that it has its own symbols: The
integers that divide a are called the divisors of a. |a |
b||means a divides b.| a does not divide b.| You might want try
your hand at proving the following basic properties which hold
for all integers a, b. c and d: Finally, suppose p is a prime
and k is a positive integer. A notation that is quickly gaining
acceptance is to write pk || a to indicate that pk divides a,
but pk+1 does not. - a | 0, 1 | a, a | a. - a | 1 if and only if
a=+/-1. - If a | b and c | d, then ac | bd. - If a | b and b |
c, then a | c. - a | b and b | a if and only if a=+/-b. - If a |
b and b is not zero, then |a| < |b|. - If a | b and a | c, then
a | (bx+cy) for all inte