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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
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 

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 ● Strategy: re 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. ● + ● 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 ● ● copy-Not distribution (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. NOT BASE wwwms Do et or 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 Ma
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 ● Strategy: re 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. ● + ● 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 ● ● copy-Not distribution (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. NOT BASE wwwms Do et or 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 Ma
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