In my number theory book there is a theorem says:

If $C$ is a complete residue system modulo $n$, $a \in \mathbb Z$ and $\gcd(a,n)=1$, So $D=\{ax+b \mid x\in C\}$ is a complete residue system modulo $n$, $\forall b\in \mathbb Z$.

My counterexample:

Let’s work mod $3$ $$C=\{0,1,2\}$$ and let $a=4$ so $\gcd(3,4)=1$ And the theorem says $\forall b\in \mathbb Z$, so we get($x=2\in C$): $$D=\{4\cdot2+0, 4\cdot2+1, 4\cdot 2+4\}=\{8,9,12\}$$ But this clearly not a comple Residue system mod $3$

Either this is a False statement (at least in a part of it) or my counterexample is False.

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    $\begingroup$ I don't understand. $x$ is meant to range through the values in $C$. $a,b$ stay fixed. Thus if $a=4, b=0$ we'd have $D=\{0,4,6\}$ which works. Not sure what you are doing in your example. $\endgroup$ – lulu Feb 23 at 16:15
  • $\begingroup$ i thought that $x$ is a constant?? $\endgroup$ – Yassir Feb 23 at 16:17
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    $\begingroup$ No. $a,b$ are constant. $x$ runs through the values of $C$. $\endgroup$ – lulu Feb 23 at 16:17
  • $\begingroup$ Is there any proof to this theorem In MSE?? $\endgroup$ – Yassir Feb 23 at 16:19
  • $\begingroup$ Prove it yourself. It's not difficult. All you have to do is to show that no two distinct elements of $D$ can be congruent $\pmod n$. $\endgroup$ – lulu Feb 23 at 16:20

$D = \{ax+b|x\in C\}$ is set builder notation. It means all elements of the form $ax+b$ where $x$ ranges over the elements in $C$.

This is similar to the set builder notation for the set of squares for example: $\{x^2|x\in \mathbb{Z}\}$

To give a concrete example, let $C=\{0, 1, 2\}$ and $a=2, b=1$ then this notation becomes: $D = \{2x+1|x\in\{0, 1, 2\}\} = \{2\cdot0+1, 2\cdot1+1, 2\cdot2+1\} = \{1, 3, 5\}$.

To prove the fact that $D$ is a complete residue system is easy. We know $\gcd(a, n)=1$ so $a$ is a unit modulo $n$. Take an multiplicative inverse $a^{-1}$ of $a$ modulo $n$. For $D$ to be a complete residue system, take any $r\in \mathbb{Z}$, we will find an $x\in C$ such that $ax+b\equiv r \mod n$. The $x$ that works is the element in $C$ congruent to $a^{-1}(r-b)$.

  • $\begingroup$ I think i have a better proof using contradiction: suppose $$ax+b\equiv ay+b \pmod n ; x,y\in C$$ So $$ax\equiv ay \pmod n \iff x\equiv y$$ because $\gcd(a,n)=1$. But this contradicts the fact that $x,y\in C$ $\endgroup$ – Yassir Feb 23 at 16:58
  • $\begingroup$ Both proofs work just fine. $\endgroup$ – jMdA Feb 23 at 17:29

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