Basic modular arithmetic question












0












$begingroup$


If for two integers $a,b$, $a=b mod x$, what is $x$ in terms of $a$ and $b$?.



I think the answer is $a-b$, but I'm not sure how to prove it without modular arithmetic, which I don't really understand.



Is there a simple way to prove and this without mods?






elementary-number-theory







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    If for two integers $a,b$, $a=b mod x$, what is $x$ in terms of $a$ and $b$?.



    I think the answer is $a-b$, but I'm not sure how to prove it without modular arithmetic, which I don't really understand.



    Is there a simple way to prove and this without mods?






    elementary-number-theory







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      If for two integers $a,b$, $a=b mod x$, what is $x$ in terms of $a$ and $b$?.



      I think the answer is $a-b$, but I'm not sure how to prove it without modular arithmetic, which I don't really understand.



      Is there a simple way to prove and this without mods?






      elementary-number-theory







      share|cite|improve this question











      $endgroup$




      If for two integers $a,b$, $a=b mod x$, what is $x$ in terms of $a$ and $b$?.



      I think the answer is $a-b$, but I'm not sure how to prove it without modular arithmetic, which I don't really understand.



      Is there a simple way to prove and this without mods?






      elementary-number-theory




      elementary-number-theory






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 7 '18 at 4:58







      James Palmer

















      asked Dec 7 '18 at 4:22









      James PalmerJames Palmer

      34




      34






















          1 Answer
          1






          active

          oldest

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          1












          $begingroup$

          This follows directly from the definition of congruence with respect to modular arithmetic (there's not much you need to know):



          $$a equiv b pmod{x} ;;; Leftrightarrow ;;; x | (a-b)$$



          (This latter expression, if you're not familiar, just means "$x$ evenly divides $(a-b)$," or, equivalently, "$(a-b)$ has no remainder when divided by $x$.")



          The problem is that, in reality, this means that, for some integer $k$,



          $$frac{a-b}{x} = k ;;; Leftrightarrow ;;; x = frac{a-b}{k}$$



          To say that $x = a-b$ (i.e. that $k=1$) means that there's some sort of context we're missing that would apply in this context. As presented, however, I think this is as far as one can get.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Use pmod{x} to get $aequiv b pmod{x}$. bmod, will produce the binary operator: bbmod x produces $bbmod x$. What you want to avoid is mod, because the spacing is wrong.
            $endgroup$
            – Arturo Magidin
            Dec 7 '18 at 4:30












          • $begingroup$
            I just was straight up using text{mod}, but noted.
            $endgroup$
            – Eevee Trainer
            Dec 7 '18 at 4:33










          • $begingroup$
            The three different modular commands are the second most common LaTeX mistake people make, I try to spread the knowledge where I can. (The most common is to use < and > instead of langle and rangle)
            $endgroup$
            – Arturo Magidin
            Dec 7 '18 at 4:34










          • $begingroup$
            Thanks! I think a better phrasing is "If two integers a,b have the same remainder when divided by x, what is x in terms of a and b?" I think I used the mod wrong, sorry!
            $endgroup$
            – James Palmer
            Dec 7 '18 at 4:35










          • $begingroup$
            I honestly just don't know much LaTeX, really. Most of what I know is self-taught or seen from experience, like using WYSIWYG editors and such. This site has at least been helpful in improving my LaTeX lol
            $endgroup$
            – Eevee Trainer
            Dec 7 '18 at 4:35











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          1 Answer
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          1 Answer
          1






          active

          oldest

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          active

          oldest

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          active

          oldest

          votes









          1












          $begingroup$

          This follows directly from the definition of congruence with respect to modular arithmetic (there's not much you need to know):



          $$a equiv b pmod{x} ;;; Leftrightarrow ;;; x | (a-b)$$



          (This latter expression, if you're not familiar, just means "$x$ evenly divides $(a-b)$," or, equivalently, "$(a-b)$ has no remainder when divided by $x$.")



          The problem is that, in reality, this means that, for some integer $k$,



          $$frac{a-b}{x} = k ;;; Leftrightarrow ;;; x = frac{a-b}{k}$$



          To say that $x = a-b$ (i.e. that $k=1$) means that there's some sort of context we're missing that would apply in this context. As presented, however, I think this is as far as one can get.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Use pmod{x} to get $aequiv b pmod{x}$. bmod, will produce the binary operator: bbmod x produces $bbmod x$. What you want to avoid is mod, because the spacing is wrong.
            $endgroup$
            – Arturo Magidin
            Dec 7 '18 at 4:30












          • $begingroup$
            I just was straight up using text{mod}, but noted.
            $endgroup$
            – Eevee Trainer
            Dec 7 '18 at 4:33










          • $begingroup$
            The three different modular commands are the second most common LaTeX mistake people make, I try to spread the knowledge where I can. (The most common is to use < and > instead of langle and rangle)
            $endgroup$
            – Arturo Magidin
            Dec 7 '18 at 4:34










          • $begingroup$
            Thanks! I think a better phrasing is "If two integers a,b have the same remainder when divided by x, what is x in terms of a and b?" I think I used the mod wrong, sorry!
            $endgroup$
            – James Palmer
            Dec 7 '18 at 4:35










          • $begingroup$
            I honestly just don't know much LaTeX, really. Most of what I know is self-taught or seen from experience, like using WYSIWYG editors and such. This site has at least been helpful in improving my LaTeX lol
            $endgroup$
            – Eevee Trainer
            Dec 7 '18 at 4:35
















          1












          $begingroup$

          This follows directly from the definition of congruence with respect to modular arithmetic (there's not much you need to know):



          $$a equiv b pmod{x} ;;; Leftrightarrow ;;; x | (a-b)$$



          (This latter expression, if you're not familiar, just means "$x$ evenly divides $(a-b)$," or, equivalently, "$(a-b)$ has no remainder when divided by $x$.")



          The problem is that, in reality, this means that, for some integer $k$,



          $$frac{a-b}{x} = k ;;; Leftrightarrow ;;; x = frac{a-b}{k}$$



          To say that $x = a-b$ (i.e. that $k=1$) means that there's some sort of context we're missing that would apply in this context. As presented, however, I think this is as far as one can get.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Use pmod{x} to get $aequiv b pmod{x}$. bmod, will produce the binary operator: bbmod x produces $bbmod x$. What you want to avoid is mod, because the spacing is wrong.
            $endgroup$
            – Arturo Magidin
            Dec 7 '18 at 4:30












          • $begingroup$
            I just was straight up using text{mod}, but noted.
            $endgroup$
            – Eevee Trainer
            Dec 7 '18 at 4:33










          • $begingroup$
            The three different modular commands are the second most common LaTeX mistake people make, I try to spread the knowledge where I can. (The most common is to use < and > instead of langle and rangle)
            $endgroup$
            – Arturo Magidin
            Dec 7 '18 at 4:34










          • $begingroup$
            Thanks! I think a better phrasing is "If two integers a,b have the same remainder when divided by x, what is x in terms of a and b?" I think I used the mod wrong, sorry!
            $endgroup$
            – James Palmer
            Dec 7 '18 at 4:35










          • $begingroup$
            I honestly just don't know much LaTeX, really. Most of what I know is self-taught or seen from experience, like using WYSIWYG editors and such. This site has at least been helpful in improving my LaTeX lol
            $endgroup$
            – Eevee Trainer
            Dec 7 '18 at 4:35














          1












          1








          1





          $begingroup$

          This follows directly from the definition of congruence with respect to modular arithmetic (there's not much you need to know):



          $$a equiv b pmod{x} ;;; Leftrightarrow ;;; x | (a-b)$$



          (This latter expression, if you're not familiar, just means "$x$ evenly divides $(a-b)$," or, equivalently, "$(a-b)$ has no remainder when divided by $x$.")



          The problem is that, in reality, this means that, for some integer $k$,



          $$frac{a-b}{x} = k ;;; Leftrightarrow ;;; x = frac{a-b}{k}$$



          To say that $x = a-b$ (i.e. that $k=1$) means that there's some sort of context we're missing that would apply in this context. As presented, however, I think this is as far as one can get.






          share|cite|improve this answer











          $endgroup$



          This follows directly from the definition of congruence with respect to modular arithmetic (there's not much you need to know):



          $$a equiv b pmod{x} ;;; Leftrightarrow ;;; x | (a-b)$$



          (This latter expression, if you're not familiar, just means "$x$ evenly divides $(a-b)$," or, equivalently, "$(a-b)$ has no remainder when divided by $x$.")



          The problem is that, in reality, this means that, for some integer $k$,



          $$frac{a-b}{x} = k ;;; Leftrightarrow ;;; x = frac{a-b}{k}$$



          To say that $x = a-b$ (i.e. that $k=1$) means that there's some sort of context we're missing that would apply in this context. As presented, however, I think this is as far as one can get.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 7 '18 at 4:33

























          answered Dec 7 '18 at 4:28









          Eevee TrainerEevee Trainer

          7,67221338




          7,67221338












          • $begingroup$
            Use pmod{x} to get $aequiv b pmod{x}$. bmod, will produce the binary operator: bbmod x produces $bbmod x$. What you want to avoid is mod, because the spacing is wrong.
            $endgroup$
            – Arturo Magidin
            Dec 7 '18 at 4:30












          • $begingroup$
            I just was straight up using text{mod}, but noted.
            $endgroup$
            – Eevee Trainer
            Dec 7 '18 at 4:33










          • $begingroup$
            The three different modular commands are the second most common LaTeX mistake people make, I try to spread the knowledge where I can. (The most common is to use < and > instead of langle and rangle)
            $endgroup$
            – Arturo Magidin
            Dec 7 '18 at 4:34










          • $begingroup$
            Thanks! I think a better phrasing is "If two integers a,b have the same remainder when divided by x, what is x in terms of a and b?" I think I used the mod wrong, sorry!
            $endgroup$
            – James Palmer
            Dec 7 '18 at 4:35










          • $begingroup$
            I honestly just don't know much LaTeX, really. Most of what I know is self-taught or seen from experience, like using WYSIWYG editors and such. This site has at least been helpful in improving my LaTeX lol
            $endgroup$
            – Eevee Trainer
            Dec 7 '18 at 4:35


















          • $begingroup$
            Use pmod{x} to get $aequiv b pmod{x}$. bmod, will produce the binary operator: bbmod x produces $bbmod x$. What you want to avoid is mod, because the spacing is wrong.
            $endgroup$
            – Arturo Magidin
            Dec 7 '18 at 4:30












          • $begingroup$
            I just was straight up using text{mod}, but noted.
            $endgroup$
            – Eevee Trainer
            Dec 7 '18 at 4:33










          • $begingroup$
            The three different modular commands are the second most common LaTeX mistake people make, I try to spread the knowledge where I can. (The most common is to use < and > instead of langle and rangle)
            $endgroup$
            – Arturo Magidin
            Dec 7 '18 at 4:34










          • $begingroup$
            Thanks! I think a better phrasing is "If two integers a,b have the same remainder when divided by x, what is x in terms of a and b?" I think I used the mod wrong, sorry!
            $endgroup$
            – James Palmer
            Dec 7 '18 at 4:35










          • $begingroup$
            I honestly just don't know much LaTeX, really. Most of what I know is self-taught or seen from experience, like using WYSIWYG editors and such. This site has at least been helpful in improving my LaTeX lol
            $endgroup$
            – Eevee Trainer
            Dec 7 '18 at 4:35
















          $begingroup$
          Use pmod{x} to get $aequiv b pmod{x}$. bmod, will produce the binary operator: bbmod x produces $bbmod x$. What you want to avoid is mod, because the spacing is wrong.
          $endgroup$
          – Arturo Magidin
          Dec 7 '18 at 4:30






          $begingroup$
          Use pmod{x} to get $aequiv b pmod{x}$. bmod, will produce the binary operator: bbmod x produces $bbmod x$. What you want to avoid is mod, because the spacing is wrong.
          $endgroup$
          – Arturo Magidin
          Dec 7 '18 at 4:30














          $begingroup$
          I just was straight up using text{mod}, but noted.
          $endgroup$
          – Eevee Trainer
          Dec 7 '18 at 4:33




          $begingroup$
          I just was straight up using text{mod}, but noted.
          $endgroup$
          – Eevee Trainer
          Dec 7 '18 at 4:33












          $begingroup$
          The three different modular commands are the second most common LaTeX mistake people make, I try to spread the knowledge where I can. (The most common is to use < and > instead of langle and rangle)
          $endgroup$
          – Arturo Magidin
          Dec 7 '18 at 4:34




          $begingroup$
          The three different modular commands are the second most common LaTeX mistake people make, I try to spread the knowledge where I can. (The most common is to use < and > instead of langle and rangle)
          $endgroup$
          – Arturo Magidin
          Dec 7 '18 at 4:34












          $begingroup$
          Thanks! I think a better phrasing is "If two integers a,b have the same remainder when divided by x, what is x in terms of a and b?" I think I used the mod wrong, sorry!
          $endgroup$
          – James Palmer
          Dec 7 '18 at 4:35




          $begingroup$
          Thanks! I think a better phrasing is "If two integers a,b have the same remainder when divided by x, what is x in terms of a and b?" I think I used the mod wrong, sorry!
          $endgroup$
          – James Palmer
          Dec 7 '18 at 4:35












          $begingroup$
          I honestly just don't know much LaTeX, really. Most of what I know is self-taught or seen from experience, like using WYSIWYG editors and such. This site has at least been helpful in improving my LaTeX lol
          $endgroup$
          – Eevee Trainer
          Dec 7 '18 at 4:35




          $begingroup$
          I honestly just don't know much LaTeX, really. Most of what I know is self-taught or seen from experience, like using WYSIWYG editors and such. This site has at least been helpful in improving my LaTeX lol
          $endgroup$
          – Eevee Trainer
          Dec 7 '18 at 4:35


















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