Our ultimate goal is to understand applications of mathematics to computers and electronic communications. Since computers speak a language of 0's and 1's, we want to develop so-called ``Boolean algebra'' or ``mod 2 arithmetic''. It turns out to be not much harder to develop ``mod m arithmetic'' and this is done in this chapter.
A lot of abstract algebra either requires properties
of the integers, or is based by analogy on the integers, or is
inspired by a property of the integers.
This is why we begin our story there.
In this chapter, we shall study divisibilty properties of
the integers, modular arithmetic, and properties of the primes.
The general idea is that we wish to learn how to solve
some simple types of equations in the integers and also
``modulo 
''. This will help us introduce finite fields later.
Along the way, we will develop some interesting
techniques, see some useful applications to
cryptography, and have some fun with more
``recreational'' topics such as calendar calculations
and the game of Nim.
The key computational methods are