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Problem 1. Let a₁,,an be n integers. We will use the notation gcd a; to denote the 1

Problem 1. Let a₁,,an be n integers. We will use the notation gcd a; to denote the 1

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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Problem 1. Let a₁,,an be n integers. We will use the notation gcd a; to denote the 1<i<n greatest common divisor of a1,, an and the notation lcm a; to denote the least common multiple of a1,..., an. 1<i<n Mimicking the proof of the attached proposition, show that: For any matrix (@ij)1<i<n,1<j<m of integers, we have lcm gcd aij 1<i<n 1<j<m ged lcm aij. 1<j<m 1<i<n Hint. What facts are used in the proof? Proposition. Let (Xij)1<i<n,1<j<m be a matrix of real numbers, then we have max min xij min max xij. 1<j<m 1<i<n 1<i<n 1<j<m Proof. Define f(i) (1 ≤ i ≤n) to be min xij. Then we have 1<j<m Therefore, we have f(i) < xij for all 1<i<n, 1<j<m. as desired. max f(i) < max Xij 1<i<n 1<i<n In particular, we have for all 1<j≤m. max f(i) < min max xij 1<j<m 1<i<n 1<i<n
Transcribed Image Text:Problem 1. Let a₁,,an be n integers. We will use the notation gcd a; to denote the 1<i<n greatest common divisor of a1,, an and the notation lcm a; to denote the least common multiple of a1,..., an. 1<i<n Mimicking the proof of the attached proposition, show that: For any matrix (@ij)1<i<n,1<j<m of integers, we have lcm gcd aij 1<i<n 1<j<m ged lcm aij. 1<j<m 1<i<n Hint. What facts are used in the proof? Proposition. Let (Xij)1<i<n,1<j<m be a matrix of real numbers, then we have max min xij min max xij. 1<j<m 1<i<n 1<i<n 1<j<m Proof. Define f(i) (1 ≤ i ≤n) to be min xij. Then we have 1<j<m Therefore, we have f(i) < xij for all 1<i<n, 1<j<m. as desired. max f(i) < max Xij 1<i<n 1<i<n In particular, we have for all 1<j≤m. max f(i) < min max xij 1<j<m 1<i<n 1<i<n
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