Please show your work,How the number of created/defined boxes is related to Pigeonhole Principal?How the number of created/defined boxes is related to given in the problem integers? **Instructions:**Read the proof below. If the numbers given in the second sentence are \(\{64, 69, 97, 118\}\), then which boxes end up with at least two numbers? (Select all)**Box Selection:**- Box 0: [ ]- Box 1: [✔]- Box 4: [ ]**Selection Result:**- Box 1: Box 1 and Box 4---**Mathematical Claim:**Among any 4 integers, there are two, \(a\) and \(b\), such that \(a^2 - b^2\) is divisible by 5.**Proof:**We create three boxes labeled \(B_0\), \(B_1\), and \(B_4\), so that integer \(n\) is placed in \(B_i\) if \(n^2 \mod 5 = i\).Given any 4 integers, when they are placed among these 3 boxes, some box (say \(B_i\)) will contain at least two integers (call them \(a\) and \(b\)) by the Pigeonhole Principle. Since \(a\) and \(b\) are both in box \(B_i\), we know that \(a^2 \mod 5 = i\) and \(b^2 \mod 5 = i\). This means \(a^2 = 5K + i\) and \(b^2 = 5Q + i\) for integers \(K\) and \(Q\). From this, it follows that \(a^2 - b^2 = 5(K - Q)\); that is, \(a^2 - b^2\) is divisible by 5.

Claim. Among any 4 integers, there are two, a and b, such that a² – 6² is divisible by 5. Proof. We create three boxes labeled B₁, B₁, and B4, so that integer n is placed in B; if n² mod 5 = i. Now given any 4 integers, when they are placed among these 3 boxes, some box (say B;) will contain at least two integers (call them a and b) by the Pigeonhole Principle. Since a and b are both in box B₂, we know that a² mod 5 i and 6² mod 5 = i. This means a² = 5K + i and b² = 5Q + ¿ for integers K and Q. From this it follows that a² − b² = 5(K – Q); that is, a²- 6² is divisible by 5. = = -

Claim. Among any 4 integers, there are two, a and b, such that a² – 6² is divisible by 5. Proof. We create three boxes labeled B₁, B₁, and B4, so that integer n is placed in B; if n² mod 5 = i. Now given any 4 integers, when they are placed among these 3 boxes, some box (say B;) will contain at least two integers (call them a and b) by the Pigeonhole Principle. Since a and b are both in box B₂, we know that a² mod 5 i and 6² mod 5 = i. This means a² = 5K + i and b² = 5Q + ¿ for integers K and Q. From this it follows that a² − b² = 5(K – Q); that is, a²- 6² is divisible by 5. = = -

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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