Mixing
Prove that $c + mathrm{lcm}(m,s)mathbb Z=(mmathbb Z+k)cap(smathbb Z+t)$ for every $c$ in $(mmathbb...
$begingroup$
The inclusion $(mmathbb{Z} + k) cap (s mathbb{Z} + t) supset c + lcm(m,s)mathbb{Z}$ is trivial, but I've been stuck with the other one for some time now. I thought about the chinese remainder theorem but couldn't actually apply it. Help?
OBS: Here we assume that the intersection is non empty.
abstract-algebra number-theory
asked Jan 28 at 19:42
Matheus AndradeMatheus Andrade
1,340418
$endgroup$
|
show 2 more comments
$begingroup$
The inclusion $(mmathbb{Z} + k) cap (s mathbb{Z} + t) supset c + lcm(m,s)mathbb{Z}$ is trivial, but I've been stuck with the other one for some time now. I thought about the chinese remainder theorem but couldn't actually apply it. Help?
OBS: Here we assume that the intersection is non empty.
abstract-algebra number-theory
asked Jan 28 at 19:42
Matheus AndradeMatheus Andrade
1,340418
$endgroup$
1
$begingroup$
@Did My original title actually had words, but someone else edited it (and I like it better now actually). Imho, excellent communication requires only that the intended meaning is conveyed clearly, correctly and without ambiguity, and I think this applies here too...
$endgroup$
– Matheus Andrade
Feb 8 at 13:35
$begingroup$
Words and clarity are not antagonistic... Many student passes by a phase when they think that symbols are more precise, then they grow up (and/or they study the masters).
$endgroup$
– Did
Feb 8 at 17:26
1
$begingroup$
@Did I agree with you in general, but in this case I think it doesn't matter. I'm quite aware symbols can often be antagonistic to clarity and then it's better to use words, but for this specific situation I think there is no loss of clarity. I admit I've already passed that phase you mention.
$endgroup$
– Matheus Andrade
Feb 8 at 17:51
$begingroup$
Hmmm... you are the one who linked words to lack of clarity, right? And I disagreed. But the link of symbols to lack of clarity was not made, so there is no need to debunk it. If you want to know, I would rather invoke ugliness here...
$endgroup$
– Did
Feb 8 at 18:10
1
$begingroup$
What I meant was that words are sufficient for clarity, but not absolutely always necessary for it. I agree it's ugly alright and I wouldn't let that title up if it was anything any more advanced, but in this case it's not. I think this discussion is meaningless since imo we seem to agree on almost everything. You can rest easy knowing I try to use as many words as I can as often as possible.
$endgroup$
– Matheus Andrade
Feb 8 at 18:17
|
show 2 more comments
$begingroup$
The inclusion $(mmathbb{Z} + k) cap (s mathbb{Z} + t) supset c + lcm(m,s)mathbb{Z}$ is trivial, but I've been stuck with the other one for some time now. I thought about the chinese remainder theorem but couldn't actually apply it. Help?
OBS: Here we assume that the intersection is non empty.
abstract-algebra number-theory
asked Jan 28 at 19:42
Matheus AndradeMatheus Andrade
1,340418
$endgroup$
The inclusion $(mmathbb{Z} + k) cap (s mathbb{Z} + t) supset c + lcm(m,s)mathbb{Z}$ is trivial, but I've been stuck with the other one for some time now. I thought about the chinese remainder theorem but couldn't actually apply it. Help?
OBS: Here we assume that the intersection is non empty.
abstract-algebra number-theory
abstract-algebra number-theory
asked Jan 28 at 19:42
Matheus AndradeMatheus Andrade
1,340418
asked Jan 28 at 19:42
Matheus AndradeMatheus Andrade
1,340418
edited Feb 8 at 21:26
Matheus Andrade
asked Jan 28 at 19:42
Matheus AndradeMatheus Andrade
1,340418
asked Jan 28 at 19:42
Matheus AndradeMatheus Andrade
1,340418
asked Jan 28 at 19:42
Matheus AndradeMatheus Andrade
1,340418
1,340418
1
$begingroup$
@Did My original title actually had words, but someone else edited it (and I like it better now actually). Imho, excellent communication requires only that the intended meaning is conveyed clearly, correctly and without ambiguity, and I think this applies here too...
$endgroup$
– Matheus Andrade
Feb 8 at 13:35
$begingroup$
Words and clarity are not antagonistic... Many student passes by a phase when they think that symbols are more precise, then they grow up (and/or they study the masters).
$endgroup$
– Did
Feb 8 at 17:26
1
$begingroup$
@Did I agree with you in general, but in this case I think it doesn't matter. I'm quite aware symbols can often be antagonistic to clarity and then it's better to use words, but for this specific situation I think there is no loss of clarity. I admit I've already passed that phase you mention.
$endgroup$
– Matheus Andrade
Feb 8 at 17:51
$begingroup$
Hmmm... you are the one who linked words to lack of clarity, right? And I disagreed. But the link of symbols to lack of clarity was not made, so there is no need to debunk it. If you want to know, I would rather invoke ugliness here...
$endgroup$
– Did
Feb 8 at 18:10
1
$begingroup$
What I meant was that words are sufficient for clarity, but not absolutely always necessary for it. I agree it's ugly alright and I wouldn't let that title up if it was anything any more advanced, but in this case it's not. I think this discussion is meaningless since imo we seem to agree on almost everything. You can rest easy knowing I try to use as many words as I can as often as possible.
$endgroup$
– Matheus Andrade
Feb 8 at 18:17
|
show 2 more comments
1
$begingroup$
@Did My original title actually had words, but someone else edited it (and I like it better now actually). Imho, excellent communication requires only that the intended meaning is conveyed clearly, correctly and without ambiguity, and I think this applies here too...
$endgroup$
– Matheus Andrade
Feb 8 at 13:35
$begingroup$
Words and clarity are not antagonistic... Many student passes by a phase when they think that symbols are more precise, then they grow up (and/or they study the masters).
$endgroup$
– Did
Feb 8 at 17:26
1
$begingroup$
@Did I agree with you in general, but in this case I think it doesn't matter. I'm quite aware symbols can often be antagonistic to clarity and then it's better to use words, but for this specific situation I think there is no loss of clarity. I admit I've already passed that phase you mention.
$endgroup$
– Matheus Andrade
Feb 8 at 17:51
$begingroup$
Hmmm... you are the one who linked words to lack of clarity, right? And I disagreed. But the link of symbols to lack of clarity was not made, so there is no need to debunk it. If you want to know, I would rather invoke ugliness here...
$endgroup$
– Did
Feb 8 at 18:10
1
$begingroup$
What I meant was that words are sufficient for clarity, but not absolutely always necessary for it. I agree it's ugly alright and I wouldn't let that title up if it was anything any more advanced, but in this case it's not. I think this discussion is meaningless since imo we seem to agree on almost everything. You can rest easy knowing I try to use as many words as I can as often as possible.
$endgroup$
– Matheus Andrade
Feb 8 at 18:17
1
1
$begingroup$
@Did My original title actually had words, but someone else edited it (and I like it better now actually). Imho, excellent communication requires only that the intended meaning is conveyed clearly, correctly and without ambiguity, and I think this applies here too...
$endgroup$
– Matheus Andrade
Feb 8 at 13:35
$begingroup$
@Did My original title actually had words, but someone else edited it (and I like it better now actually). Imho, excellent communication requires only that the intended meaning is conveyed clearly, correctly and without ambiguity, and I think this applies here too...
$endgroup$
– Matheus Andrade
Feb 8 at 13:35
$begingroup$
Words and clarity are not antagonistic... Many student passes by a phase when they think that symbols are more precise, then they grow up (and/or they study the masters).
$endgroup$
– Did
Feb 8 at 17:26
$begingroup$
Words and clarity are not antagonistic... Many student passes by a phase when they think that symbols are more precise, then they grow up (and/or they study the masters).
$endgroup$
– Did
Feb 8 at 17:26
1
1
$begingroup$
@Did I agree with you in general, but in this case I think it doesn't matter. I'm quite aware symbols can often be antagonistic to clarity and then it's better to use words, but for this specific situation I think there is no loss of clarity. I admit I've already passed that phase you mention.
$endgroup$
– Matheus Andrade
Feb 8 at 17:51
$begingroup$
@Did I agree with you in general, but in this case I think it doesn't matter. I'm quite aware symbols can often be antagonistic to clarity and then it's better to use words, but for this specific situation I think there is no loss of clarity. I admit I've already passed that phase you mention.
$endgroup$
– Matheus Andrade
Feb 8 at 17:51
$begingroup$
Hmmm... you are the one who linked words to lack of clarity, right? And I disagreed. But the link of symbols to lack of clarity was not made, so there is no need to debunk it. If you want to know, I would rather invoke ugliness here...
$endgroup$
– Did
Feb 8 at 18:10
$begingroup$
Hmmm... you are the one who linked words to lack of clarity, right? And I disagreed. But the link of symbols to lack of clarity was not made, so there is no need to debunk it. If you want to know, I would rather invoke ugliness here...
$endgroup$
– Did
Feb 8 at 18:10
1
1
$begingroup$
What I meant was that words are sufficient for clarity, but not absolutely always necessary for it. I agree it's ugly alright and I wouldn't let that title up if it was anything any more advanced, but in this case it's not. I think this discussion is meaningless since imo we seem to agree on almost everything. You can rest easy knowing I try to use as many words as I can as often as possible.
$endgroup$
– Matheus Andrade
Feb 8 at 18:17
$begingroup$
What I meant was that words are sufficient for clarity, but not absolutely always necessary for it. I agree it's ugly alright and I wouldn't let that title up if it was anything any more advanced, but in this case it's not. I think this discussion is meaningless since imo we seem to agree on almost everything. You can rest easy knowing I try to use as many words as I can as often as possible.
$endgroup$
– Matheus Andrade
Feb 8 at 18:17
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Hint $,c',cin (k+mBbb{Z}) cap (t+s Bbb{Z}),Rightarrow,c'-cin mBbb Z,sBbb Z,Rightarrow,c'-cin mBbb Zcap nBbb Z = {rm lcm}(m,n)Bbb Z$
Remark $ $ This is equivalent to the uniqueness of a solution $,x = c,$ of the following congruences
$$begin{align} x&equiv kpmod{m}\ x&equiv tpmod{s}end{align}$$
If $,x = c'$ is another solution then $, c'equiv xequiv cpmod{! m},$ so $ mmid c'-c.,$ Similarly $,smid c'-c,$ therefore $,ell := {rm lcm}(m,s)mid c'-c,,$ thus $,c'equiv cpmod{!ell},,$ i.e. any solution is unique $!bmod ell.,$ Therefore if you know that form of CRT then it follows immediately from the uniqueness part.
answered Jan 28 at 20:26
Bill DubuqueBill Dubuque
213k29195654
$endgroup$
add a comment |
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$begingroup$
Hint $,c',cin (k+mBbb{Z}) cap (t+s Bbb{Z}),Rightarrow,c'-cin mBbb Z,sBbb Z,Rightarrow,c'-cin mBbb Zcap nBbb Z = {rm lcm}(m,n)Bbb Z$
Remark $ $ This is equivalent to the uniqueness of a solution $,x = c,$ of the following congruences
$$begin{align} x&equiv kpmod{m}\ x&equiv tpmod{s}end{align}$$
If $,x = c'$ is another solution then $, c'equiv xequiv cpmod{! m},$ so $ mmid c'-c.,$ Similarly $,smid c'-c,$ therefore $,ell := {rm lcm}(m,s)mid c'-c,,$ thus $,c'equiv cpmod{!ell},,$ i.e. any solution is unique $!bmod ell.,$ Therefore if you know that form of CRT then it follows immediately from the uniqueness part.
answered Jan 28 at 20:26
Bill DubuqueBill Dubuque
213k29195654
$endgroup$
add a comment |
$begingroup$
Hint $,c',cin (k+mBbb{Z}) cap (t+s Bbb{Z}),Rightarrow,c'-cin mBbb Z,sBbb Z,Rightarrow,c'-cin mBbb Zcap nBbb Z = {rm lcm}(m,n)Bbb Z$
Remark $ $ This is equivalent to the uniqueness of a solution $,x = c,$ of the following congruences
$$begin{align} x&equiv kpmod{m}\ x&equiv tpmod{s}end{align}$$
If $,x = c'$ is another solution then $, c'equiv xequiv cpmod{! m},$ so $ mmid c'-c.,$ Similarly $,smid c'-c,$ therefore $,ell := {rm lcm}(m,s)mid c'-c,,$ thus $,c'equiv cpmod{!ell},,$ i.e. any solution is unique $!bmod ell.,$ Therefore if you know that form of CRT then it follows immediately from the uniqueness part.
answered Jan 28 at 20:26
Bill DubuqueBill Dubuque
213k29195654
$endgroup$
add a comment |
$begingroup$
Hint $,c',cin (k+mBbb{Z}) cap (t+s Bbb{Z}),Rightarrow,c'-cin mBbb Z,sBbb Z,Rightarrow,c'-cin mBbb Zcap nBbb Z = {rm lcm}(m,n)Bbb Z$
Remark $ $ This is equivalent to the uniqueness of a solution $,x = c,$ of the following congruences
$$begin{align} x&equiv kpmod{m}\ x&equiv tpmod{s}end{align}$$
If $,x = c'$ is another solution then $, c'equiv xequiv cpmod{! m},$ so $ mmid c'-c.,$ Similarly $,smid c'-c,$ therefore $,ell := {rm lcm}(m,s)mid c'-c,,$ thus $,c'equiv cpmod{!ell},,$ i.e. any solution is unique $!bmod ell.,$ Therefore if you know that form of CRT then it follows immediately from the uniqueness part.
answered Jan 28 at 20:26
Bill DubuqueBill Dubuque
213k29195654
$endgroup$
Hint $,c',cin (k+mBbb{Z}) cap (t+s Bbb{Z}),Rightarrow,c'-cin mBbb Z,sBbb Z,Rightarrow,c'-cin mBbb Zcap nBbb Z = {rm lcm}(m,n)Bbb Z$
Remark $ $ This is equivalent to the uniqueness of a solution $,x = c,$ of the following congruences
$$begin{align} x&equiv kpmod{m}\ x&equiv tpmod{s}end{align}$$
If $,x = c'$ is another solution then $, c'equiv xequiv cpmod{! m},$ so $ mmid c'-c.,$ Similarly $,smid c'-c,$ therefore $,ell := {rm lcm}(m,s)mid c'-c,,$ thus $,c'equiv cpmod{!ell},,$ i.e. any solution is unique $!bmod ell.,$ Therefore if you know that form of CRT then it follows immediately from the uniqueness part.
answered Jan 28 at 20:26
Bill DubuqueBill Dubuque
213k29195654
edited Jan 28 at 20:50
answered Jan 28 at 20:26
Bill DubuqueBill Dubuque
213k29195654
answered Jan 28 at 20:26
Bill DubuqueBill Dubuque
213k29195654
answered Jan 28 at 20:26
Bill DubuqueBill Dubuque
213k29195654
213k29195654
add a comment |
add a comment |
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$begingroup$
@Did My original title actually had words, but someone else edited it (and I like it better now actually). Imho, excellent communication requires only that the intended meaning is conveyed clearly, correctly and without ambiguity, and I think this applies here too...
$endgroup$
– Matheus Andrade
Feb 8 at 13:35
$begingroup$
Words and clarity are not antagonistic... Many student passes by a phase when they think that symbols are more precise, then they grow up (and/or they study the masters).
$endgroup$
– Did
Feb 8 at 17:26
1
$begingroup$
@Did I agree with you in general, but in this case I think it doesn't matter. I'm quite aware symbols can often be antagonistic to clarity and then it's better to use words, but for this specific situation I think there is no loss of clarity. I admit I've already passed that phase you mention.
$endgroup$
– Matheus Andrade
Feb 8 at 17:51
$begingroup$
Hmmm... you are the one who linked words to lack of clarity, right? And I disagreed. But the link of symbols to lack of clarity was not made, so there is no need to debunk it. If you want to know, I would rather invoke ugliness here...
$endgroup$
– Did
Feb 8 at 18:10
1
$begingroup$
What I meant was that words are sufficient for clarity, but not absolutely always necessary for it. I agree it's ugly alright and I wouldn't let that title up if it was anything any more advanced, but in this case it's not. I think this discussion is meaningless since imo we seem to agree on almost everything. You can rest easy knowing I try to use as many words as I can as often as possible.
$endgroup$
– Matheus Andrade
Feb 8 at 18:17