Number Theory Section Summary: 2.1

The Division Algorithm

``...the foundation stone upon which our whole development rests.'' (p. 17)

1. Theorems

   Division Algorithm: Given integers a and b, with b>0, there exist unique integers q and r satisfying

   

   with . q is called the quotient, and r is called the remainder.

   If a>0 as well, then this is an obvious extension of the Archimedean property: if any positive b can be added to itself enough times to exceed any positive a, then clearly there will come a point at which and (q+1)b > a. r just represents the amount by which qb is short (if any!).

   (Proof using well-ordering and contradiction.)

   Corollary: Given integers a and b, with , there exist unique integers q and r satisfying

   

   with .

2. Notes

   The author shows a couple of interesting properties immediately:

   • b=2 leads to the definition of even and odd numbers, as 2q or 2q+1.
   • Furthermore, every square of an integer is of the form 4k or 4k+1.
   • b=4 leads to the conclusion that every square of an odd is of the form 8k+1.

3. Summary

   Burton comments that the focus will fall on the applications of the division algorithm: ``...it allows us to prove assertions about all the integers by considering only a finite number of cases.'' (p. 19)