text: 1. Inverse Modulus
  
  From the definition, it says that Let $\displaystyle m\in\mathbb{Z^{+}}$ and $\displaystyle a\in\mathbb{Z}$. An integer $\displaystyle \overline{a}$ such that $\displaystyle \overline{a}a\equiv1\quad(mod\; m)$ is an Inverse of $\displaystyle a\; mod\; m$
  
  From how I interpret it, it is...
  
  $\displaystyle a\; mod\; m\;=$Remainder of $\displaystyle a\div m$
  
  Inverse of $\displaystyle a\; mod\; m\;=a-m\cdot k$, where $\displaystyle k\in\mathbb{Z}$. Is this right?
  
  So if I have $\displaystyle 3\; mod\;5$, The normal $\displaystyle 3\; mod\;5=3$. Inverse $\displaystyle 3\; mod\;5=-2$
  
  Then it says that $\displaystyle \overline{a}a\equiv1\quad(mod\; m)$, does it mean $\displaystyle -2\cdot3\equiv1\quad(mod\; m)$ and so when $\displaystyle (-2\times3)\div3$, the remainder is 1? But it doesn't look like it.
  
  I am very confused with the notation and how this whole inverse of modulus work.
  
  2. Hello, xEnOn!
  
  There are two inverses: additive and multiplicative.
  You are confusing them.
  
  Here is a corrected version of what you said . . .
  
  $\displaystyle \text{From the d{e}finition, it says that: }\,\text{Let }m\in\mathbb{Z^{+}}\,\text{ and }\,a\in\mathbb{Z}.$
  
  $\displaystyle \text{An integer }\overline{a}\text{ such that }\overline{a}\!\cdot\!a\equiv1\text{ (mod }m)$
  
  . . $\displaystyle \text{is a }multiplicative\text{ inverse of }a\text{ (mod }m)$
  
  3. Oh... But what's the difference between additive and multiplicative?
  Was my interpretation additive or multiplicative and was it correct in the first place? Because I actually find my own intepretation pretty confusing too.
explanation: The text extract is a discussion about the concept of the inverse in modular arithmetic, specifically focusing on the multiplicative inverse. The user is confused about the definition and application of the multiplicative inverse and asks for clarification. The text also includes a response from another user (xEnOn) who attempts to correct and explain the difference between additive and multiplicative inverses.
  
  The extract contains a question and a partial answer, but the user's confusion and additional questions indicate that the answer is not complete.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 1. Inverse Modulus
  
  From the definition, it says that Let $\displaystyle m\in\mathbb{Z^{+}}$ and $\displaystyle a\in\mathbb{Z}$. An integer $\displaystyle \overline{a}$ such that $\displaystyle \overline{a}a\equiv1\quad(mod\; m)$ is an Inverse of $\displaystyle a\; mod\; m$
  
  From how I interpret it, it is...
  
  $\displaystyle a\; mod\; m\;=$Remainder of $\displaystyle a\div m$
  
  Inverse of $\displaystyle a\; mod\; m\;=a-m\cdot k$, where $\displaystyle k\in\mathbb{Z}$. Is this right?
  
  So if I have $\displaystyle 3\; mod\;5$, The normal $\displaystyle 3\; mod\;5=3$. Inverse $\displaystyle 3\; mod\;5=-2$
  
  Then it says that $\displaystyle \overline{a}a\equiv1\quad(mod\; m)$, does it mean $\displaystyle -2\cdot3\equiv1\quad(mod\; m)$ and so when $\displaystyle (-2\times3)\div3$, the remainder is 1? But it doesn't look like it.
  
  I am very confused with the notation and how this whole inverse of modulus work.
  
  2. Hello, xEnOn!
  
  There are two inverses: additive and multiplicative.
  You are confusing them.
  
  Here is a corrected version of what you said . . .
  
  $\displaystyle \text{From the d{e}finition, it says that: }\,\text{Let }m\in\mathbb{Z^{+}}\,\text{ and }\,a\in\mathbb{Z}.$
  
  $\displaystyle \text{An integer }\overline{a}\text{ such that }\overline{a}\!\cdot\!a\equiv1\text{ (mod }m)$
  
  . . $\displaystyle \text{is a }multiplicative\text{ inverse of }a\text{ (mod }m)$
  
  3. Oh... But what's the difference between additive and multiplicative?
  Was my interpretation additive or multiplicative and was it correct in the first place? Because I actually find my own intepretation pretty confusing too.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is a discussion about the concept of the inverse in modular arithmetic, specifically focusing on the multiplicative inverse. The user is confused about the definition and application of the multiplicative inverse and asks for clarification. The text also includes a response from another user (xEnOn) who attempts to correct and explain the difference between additive and multiplicative inverses.
  
  The extract contains a question and a partial answer, but the user's confusion and additional questions indicate that the answer is not complete.
  
  Label: question-and-answer