Prove this claim: For every positive integer d and every integer n, the following equation is satisfied: (n²) div d = (d)( n div d)² + (2)( n div d )( n mod d) + (((n mod d)²) div d)
Prove this claim: For every positive integer d and every integer n, the following equation is satisfied: (n²) div d = (d)( n div d)² + (2)( n div d )( n mod d) + (((n mod d)²) div d)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove this claim by follow the below all strategy. You can take your time but please do it correctly and handwritten
Transcribed Image Text:Prove this claim:
Do not copy
For every positive integer d and every integer n, the following equation is satisfied:
(n² ) div d = (d)( n div d)² + ( 2 )( n div d)( n mod d) + (((n mod d)²) div d)
for distru
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Strategy:
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Begin your proof by supposing that d is a positive integer and n is an integer (and do not assume anything more).
Based on that limited information you must establish the correctness of the asserted equation.
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copy No
You can assume that the reader accepts the truth of the Quotient-Remainder Theorem, but they might not
remember the definitions of the "div" and "mod" operators associated with integer division; reminders are in
order. Take care of that before launching into the gory details of your argument.
State that the (ordinary) Quotient-Remainder Theorem justifies these statements concerning the existence and
et couf uniqueness of certain quotients and remainders obtained through integer division using the divisor d:
ution-Do
ions Do
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(1) 31 (9₁, ₁) = ZxZ, (n=dq₁ + ₁)^(0 ≤r₁ <d)
(2) 3! (92,7₂) € ZXZ, ((r₁)² = dq2 + r₂)^(0 ≤ r₂<d)
(3) 31 (93,73) € ZXZ, (n² = dq3 +r3) ^ (0≤r³<d)
Identify 93, 91, r₁, and q2 with the appropriate div and mod values that appear in the equation that you want to
prove is true.
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Prove that q3 can be determined using d, q1, r1, and 92:
o Accomplish this by using a substitution, some algebra, and another substitution, to rewrite n² in terms of
the quantities d, q1, r1, 92, and r2. The uniqueness of the ordered pair (93, r3) that satisfies the condition
(n² = dq3+13)^(0 ≤ 3<d) will allow you to equate a certain expression with q3.
Explain why the relationship between q3 and d, 91, r₁, and q2 is a demonstration of what was to be proved.
Do not co
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