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How to find a reduced residue system modulo of a number?

Show that if $c_1, c_2, ldots, c_phi(m)$ is a reduced residue system modulo m, $m neq 2$ then $c_1 + cdots+ c_phi(m) equiv 0 pmodm$Symmetry in reduced residue systemsCounting elements of reduced residue systems modulo one number which are smaller than anotherMeasuring the biggest difference in the reduced residue system modulo NQuestion about the reduced residue system for a given primorialProve that $1^n,2^n,3^n,ldots,(p-1)^n$ is a reduced residue system modulo $p$Question about congruence classes and reduced residue systemsWhich of the following are reduced modulo residue systems modulo 18?$r_1,r_2,…,r_phi(m)$ is a reduced residue system modulo $m$ iff $r_1+k,r_2+k,…,r_phi(m)+k$ be a reduced residue system modulo $m$Explanation about reduced residue system theorem

1

$begingroup$

The reduced residue system modulo $10$ is: $1, 3, 7, 9$

But how could we find these numbers?

The only thing I know is they're relatively prime to $10$.
What does it mean by "no two different elements of the set are congruent to modulo m"?

Thanks,

Chan

elementary-number-theory

share|cite|improve this question

asked Feb 26 '11 at 23:23

ChanChan

5,4361179123

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  If you know that they are relatively prime to 10, isn't that your answer?
  $endgroup$
  – PrimeNumber
  Feb 26 '11 at 23:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @PEV: Thanks, I understand it now. How about the second part "no two different elements of the set are congruent to modulo m". Could you help me explain this as well?
  $endgroup$
  – Chan
  Feb 26 '11 at 23:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Exactly. the first part gives the solution while the second part tell when to stop.
  $endgroup$
  – Guest
  Mar 27 '14 at 7:53
  

  
  
  
  

add a comment | 

1

$begingroup$

The reduced residue system modulo $10$ is: $1, 3, 7, 9$

But how could we find these numbers?

The only thing I know is they're relatively prime to $10$.
What does it mean by "no two different elements of the set are congruent to modulo m"?

Thanks,

Chan

elementary-number-theory

share|cite|improve this question

asked Feb 26 '11 at 23:23

ChanChan

5,4361179123

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  If you know that they are relatively prime to 10, isn't that your answer?
  $endgroup$
  – PrimeNumber
  Feb 26 '11 at 23:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @PEV: Thanks, I understand it now. How about the second part "no two different elements of the set are congruent to modulo m". Could you help me explain this as well?
  $endgroup$
  – Chan
  Feb 26 '11 at 23:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Exactly. the first part gives the solution while the second part tell when to stop.
  $endgroup$
  – Guest
  Mar 27 '14 at 7:53
  

  
  
  
  

add a comment | 

$begingroup$

The reduced residue system modulo $10$ is: $1, 3, 7, 9$

But how could we find these numbers?

The only thing I know is they're relatively prime to $10$.
What does it mean by "no two different elements of the set are congruent to modulo m"?

Thanks,

Chan

elementary-number-theory

share|cite|improve this question

asked Feb 26 '11 at 23:23

ChanChan

5,4361179123

$endgroup$

share|cite|improve this question

asked Feb 26 '11 at 23:23

ChanChan

5,4361179123

share|cite|improve this question

asked Feb 26 '11 at 23:23

ChanChan

5,4361179123

asked Feb 26 '11 at 23:23

ChanChan

5,4361179123

asked Feb 26 '11 at 23:23

ChanChan

5,4361179123

2 Answers
2

active

oldest

votes

5

$begingroup$

The line "no two different elements of the set are congruent modulo $m$" just means that all of your elements are distinct modulo $m$. For example, $1,3,7,9,11,111,1111$ are all relatively prime to $10$, but they do not form a reduced residue system since $1,11,111,1111$ are all the same modulo $10$

Another way to specify the condition is: The reduced residue system modulo $N$ is the set of all integers $m$ with $gcd(m,N)=1$ and $0leq mleq N$.

Hope that helps,

share|cite|improve this answer

answered Feb 26 '11 at 23:32

Eric NaslundEric Naslund

60.6k10140241

$endgroup$

add a comment | 

1

$begingroup$

Presumably you refer to the Wikipedia definition of reduced residue system. The point of the definition is to specify a system of representatives for the $rm:phi(n):$ congruence classes that are units (invertible) $rm (mod n):.:$ This amounts to choosing a set of $rm:phi(n):$ integers coprime to $rm:n:$ such they they are all distinct $rm (mod m):$.

share|cite|improve this answer

edited Oct 1 '17 at 20:37

davidlowryduda♦

75k7120256

answered Feb 26 '11 at 23:52

Bill DubuqueBill Dubuque

212k29195654

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  How on earth do you think this answer is better. This is unclear at best unless you already understand the material.
  $endgroup$
  – Eric Naslund
  Feb 28 '11 at 16:06
  

  
  
  
  

add a comment | 

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5

$begingroup$

The line "no two different elements of the set are congruent modulo $m$" just means that all of your elements are distinct modulo $m$. For example, $1,3,7,9,11,111,1111$ are all relatively prime to $10$, but they do not form a reduced residue system since $1,11,111,1111$ are all the same modulo $10$

Another way to specify the condition is: The reduced residue system modulo $N$ is the set of all integers $m$ with $gcd(m,N)=1$ and $0leq mleq N$.

Hope that helps,

share|cite|improve this answer

answered Feb 26 '11 at 23:32

Eric NaslundEric Naslund

60.6k10140241

$endgroup$

add a comment | 

5

$begingroup$

The line "no two different elements of the set are congruent modulo $m$" just means that all of your elements are distinct modulo $m$. For example, $1,3,7,9,11,111,1111$ are all relatively prime to $10$, but they do not form a reduced residue system since $1,11,111,1111$ are all the same modulo $10$

Another way to specify the condition is: The reduced residue system modulo $N$ is the set of all integers $m$ with $gcd(m,N)=1$ and $0leq mleq N$.

Hope that helps,

share|cite|improve this answer

answered Feb 26 '11 at 23:32

Eric NaslundEric Naslund

60.6k10140241

$endgroup$

add a comment | 

$begingroup$

The line "no two different elements of the set are congruent modulo $m$" just means that all of your elements are distinct modulo $m$. For example, $1,3,7,9,11,111,1111$ are all relatively prime to $10$, but they do not form a reduced residue system since $1,11,111,1111$ are all the same modulo $10$

Another way to specify the condition is: The reduced residue system modulo $N$ is the set of all integers $m$ with $gcd(m,N)=1$ and $0leq mleq N$.

Hope that helps,

share|cite|improve this answer

answered Feb 26 '11 at 23:32

Eric NaslundEric Naslund

60.6k10140241

$endgroup$

share|cite|improve this answer

answered Feb 26 '11 at 23:32

Eric NaslundEric Naslund

60.6k10140241

answered Feb 26 '11 at 23:32

Eric NaslundEric Naslund

60.6k10140241

answered Feb 26 '11 at 23:32

Eric NaslundEric Naslund

60.6k10140241

1

$begingroup$

Presumably you refer to the Wikipedia definition of reduced residue system. The point of the definition is to specify a system of representatives for the $rm:phi(n):$ congruence classes that are units (invertible) $rm (mod n):.:$ This amounts to choosing a set of $rm:phi(n):$ integers coprime to $rm:n:$ such they they are all distinct $rm (mod m):$.

share|cite|improve this answer

edited Oct 1 '17 at 20:37

davidlowryduda♦

75k7120256

answered Feb 26 '11 at 23:52

Bill DubuqueBill Dubuque

212k29195654

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  How on earth do you think this answer is better. This is unclear at best unless you already understand the material.
  $endgroup$
  – Eric Naslund
  Feb 28 '11 at 16:06
  

  
  
  
  

add a comment | 

1

$begingroup$

Presumably you refer to the Wikipedia definition of reduced residue system. The point of the definition is to specify a system of representatives for the $rm:phi(n):$ congruence classes that are units (invertible) $rm (mod n):.:$ This amounts to choosing a set of $rm:phi(n):$ integers coprime to $rm:n:$ such they they are all distinct $rm (mod m):$.

share|cite|improve this answer

edited Oct 1 '17 at 20:37

davidlowryduda♦

75k7120256

answered Feb 26 '11 at 23:52

Bill DubuqueBill Dubuque

212k29195654

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  How on earth do you think this answer is better. This is unclear at best unless you already understand the material.
  $endgroup$
  – Eric Naslund
  Feb 28 '11 at 16:06
  

  
  
  
  

add a comment | 

$begingroup$

Presumably you refer to the Wikipedia definition of reduced residue system. The point of the definition is to specify a system of representatives for the $rm:phi(n):$ congruence classes that are units (invertible) $rm (mod n):.:$ This amounts to choosing a set of $rm:phi(n):$ integers coprime to $rm:n:$ such they they are all distinct $rm (mod m):$.

share|cite|improve this answer

edited Oct 1 '17 at 20:37

davidlowryduda♦

75k7120256

answered Feb 26 '11 at 23:52

Bill DubuqueBill Dubuque

212k29195654

$endgroup$

share|cite|improve this answer

edited Oct 1 '17 at 20:37

davidlowryduda♦

75k7120256

answered Feb 26 '11 at 23:52

Bill DubuqueBill Dubuque

212k29195654

edited Oct 1 '17 at 20:37

davidlowryduda♦

75k7120256

edited Oct 1 '17 at 20:37

davidlowryduda♦

75k7120256

answered Feb 26 '11 at 23:52

Bill DubuqueBill Dubuque

212k29195654

answered Feb 26 '11 at 23:52

Bill DubuqueBill Dubuque

212k29195654