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

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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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