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The quotient-remainder theorem says not only that there
Chapter 4, Problem 18E(choose chapter or problem)
The quotient-remainder theorem says not only that there exist quotients and remainders but also that the quotient and remainder of a division are unique. Prove the uniqueness. That is, prove that if a and d are integers with d > 0 and if \(q_{1}, r_{1}, q_{2}, \text { and } r_{2}\) are integers such that
\(a=d q_{1}+r_{1} \quad \text { where } 0 \leq r_{1}<d\)
and
\(a=d q_{2}+r_{2} \quad \text { where } 0 \leq r_{2}<d\)
then
\(q_{1}=q_{2} \quad \text { and } \quad r_{1}=r_{2}\)
Text Transcription:
q_1, r_1, q_2, and r_2
a = dq_1 + r_1 where 0 leq r_1 < d
a = dq_2 + r_2 where 0 leq r_2 < d
q_1 = q_2 and r_1 = r_2
Questions & Answers
QUESTION:
The quotient-remainder theorem says not only that there exist quotients and remainders but also that the quotient and remainder of a division are unique. Prove the uniqueness. That is, prove that if a and d are integers with d > 0 and if \(q_{1}, r_{1}, q_{2}, \text { and } r_{2}\) are integers such that
\(a=d q_{1}+r_{1} \quad \text { where } 0 \leq r_{1}<d\)
and
\(a=d q_{2}+r_{2} \quad \text { where } 0 \leq r_{2}<d\)
then
\(q_{1}=q_{2} \quad \text { and } \quad r_{1}=r_{2}\)
Text Transcription:
q_1, r_1, q_2, and r_2
a = dq_1 + r_1 where 0 leq r_1 < d
a = dq_2 + r_2 where 0 leq r_2 < d
q_1 = q_2 and r_1 = r_2
ANSWER:Solution:
Step 1:
In this question we have to prove that if a and d are integers with d > 0 and if ,
,
, and
are integers such that
where 0 ≤
<d and
where
0<<d, then
=
and
=
..