Understanding the given proof of integers - Ring theory

In summary: It is the same method that is usually used to show that two sets ##A## and ##B## are equal. We show ##A\subseteq B## (##a\in A \Longrightarrow a\in B##) and ##B\subseteq A## (##b\in B \Longrightarrow b\in A##) and conclude ##A=B.
  • #1
chwala
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Homework Statement
see attached
Relevant Equations
Ring Theory
My interest is on the highlighted part ...

1691205092323.png


1691205125883.png


Now to my question,

in what cases do we have ##mn<(m,n)[m,n]?##

I was able to use my example say,
Let ##m=24## and ##n=30## for example, then
##[m,n]=120## and ##(m,n)=6## in this case we can verify that,
##720=6⋅120## implying that, ##mn≤ (m,n)[m,n]##.
 
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  • #2
chwala said:
Now to my question,

in what cases do we have ##mn<(m,n)[m,n]?##
This is a strange question, in the very next line the finish the proof, that it is an equality.
 
  • #3
martinbn said:
This is a strange question, in the very next line the finish the proof, that it is an equality.
I get your point the last line indicates an equal sign. However, ...the preceding line states that,
"Therefore, it must be less than the greatest common divisor'... on the contrary should it not be 'Therefore, it is equal to the greatest common divisor'? Unless there are cases where the inequality applies.
 
  • #4
chwala said:
Homework Statement: see attached
Relevant Equations: Ring Theory

My interest is on the highlighted part ...

View attachment 330141

View attachment 330142

Now to my question,

in what cases do we have ##mn<(m,n)[m,n]?##
Never. We have ##\geq## and ##\leq## making it ##=## and completing the proof.

chwala said:
I was able to use my example say,
Let ##m=24## and ##n=30## for example, then
##[m,n]=120## and ##(m,n)=6## in this case we can verify that,
##720=6⋅120## implying that, ##mn≤ (m,n)[m,n]##.
The location with your red mark comes from ##a\leq b \Longrightarrow a\cdot c\leq b\cdot c## in case ##c\geq 0.## With ##a=\dfrac{mn}{[m,n]}\, , \,b=(m,n)## and ##c=[m,n]## we get what is written.
 
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  • #5
fresh_42 said:
Never. We have ##\geq## and ##\leq## making it ##=## and completing the proof.The location with your red mark comes from ##a\leq b \Longrightarrow a\cdot c\leq b\cdot c## in case ##c\geq 0.## With ##a=\dfrac{mn}{[m,n]}\, , \,b=(m,n)## and ##c=[m,n]## we get what is written.
I can now see that two proofs that involve the inequalities ##[≤]## and ##[≥]## in general imply ##[=]##, thus concluding the proof. Clear now...
 
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  • #6
chwala said:
I can now see that two proofs that involve the inequalities ##[≤]## and ##[≥]## in general imply ##[=]##, thus concluding the proof. Clear now...
This is a standard way of proving that two quantities are equal. If you can show that ##a \le b## and that ##a \ge b##, then you can conclude that a = b.
 
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  • #7
chwala said:
I can now see that two proofs that involve the inequalities ##[≤]## and ##[≥]## in general imply ##[=]##, thus concluding the proof. Clear now...
It is the same method that is usually used to show that two sets ##A## and ##B## are equal. We show ##A\subseteq B## (##a\in A \Longrightarrow a\in B##) and ##B\subseteq A## (##b\in B \Longrightarrow b\in A##) and conclude ##A=B.##
 
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