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.]
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.© BrainMass Inc. brainmass.com March 4, 2021, 6:43 pm ad1c9bdddf
Please see the attached file for the complete solution.
Thanks for using BrainMass.
Let and be positive integers and
(a) Show that for all integers .
Since . From the condition, , so . is always true. Thus , for all integers ...
The Division Theorem is investigated. The solution is detailed and well presented.