# The Division Theorem

Let p and a be positive integers and suppose that p|a2.

a) Show that p|(ra + sp)2 for all integers r; s.

b) Use part a), the definition of prime integer, and Theorem 15.1.1 to

construct a proof by induction that p|a. [Hint: If a (< or =) p consider

p = qa + r, where 0 (< or =) r < a. If p < a consider a = qp + r, where

0 (< or =) r < p.]

Theorem 15.1.1:

The division theorem

Let a and b be integers with b> 0. Then there are unique integers q and r such that a= bq+r and 0(< or =) r < b.

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Let and be positive integers and

(a) Show that for all integers .

Proof:

Since . From the condition, , so . is always true. Thus , for all integers ...

#### Solution Summary

The Division Theorem is investigated. The solution is detailed and well presented.