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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help me proof this without using complicated proofs.
Thank you
![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 ai
1<i<n 1<j<m
gcd lem 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F029f6964-db88-4f29-818c-83a6e31791c3%2Fb7a20933-c802-40c1-8725-93df42b9cf64%2Fh0h6jgo_processed.png&w=3840&q=75)
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 ai
1<i<n 1<j<m
gcd lem 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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