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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