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In the following statement, it seems "mod" serves two separate functions.

$(ab)\textrm{ mod } n \equiv \big((a\textrm{ mod } n)\cdot(b\textrm{ mod } n)\big)\textrm{ mod } n$

  • There is the use as a qualification of a statement about equivalence, as in the final "mod n": "these two expressions (LHS and RHS of $\equiv$) are equivalent mod n."
  • There is the use as an operator applied to a single expression, as in all the other occurences.

Is there any ambiguity here? Any potential mathematical hazards that can arise from not being clear about the distinction?

EDIT - thinking on, I can see that the final "mod n" could be interpreted in either sense...)

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    $\begingroup$ There's a difference between the way mathematicians and programmers use "mod". The former refers to an equivalence class, the latter to a remainder (the least positive member of the equivalence class). Both usages are fine, but the difference between them can cause some confusion. $\endgroup$
    – lulu
    Oct 23, 2021 at 15:49
  • $\begingroup$ So what about the mixture in the statement in the question - is it correct by the standards of either camp? $\endgroup$
    – Robin Andrews
    Oct 23, 2021 at 15:52
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    $\begingroup$ The statement I would prefer is that if $a\equiv A \pmod n$ and $b\equiv B\pmod n$ then $ab\equiv AB \pmod n$. good enough? $\endgroup$
    – lulu
    Oct 23, 2021 at 15:54
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    $\begingroup$ Nevertheless, your statement is also correct. (Follows by Lulu's comment.) $\endgroup$
    – Berci
    Oct 23, 2021 at 16:04
  • $\begingroup$ Wow, that was impressive for you to accurately locate and articulate that difference! $\endgroup$
    – Trebor
    Oct 23, 2021 at 16:14

1 Answer 1

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It is customary to write the equivalence relation as $a \equiv b \pmod m$, with parenthesis around $\bmod m$ and a bigger space between $a \equiv b$ and $\bmod m$.

Even $\TeX$ (and $\LaTeX$ and MathJax) do that with the appropiate commands:

The equivalence relation a \equiv b \pmod n renders as $a \equiv b \pmod n$

The binary operation a = b \bmod n renders as $a = b \bmod n$

That usually suffices to avoid ambiguity since it wouldn't make sense to put parenthesis like that in the $\bmod$ operation.

The expression you have probably should be written like

$$(a b) \bmod n \equiv ((a \bmod n)\cdot(b \bmod n)) \pmod n$$

if the intention is that the three first $\bmod$'s denote the operation and the last one denotes the equivalence relation or

$$(a b) \bmod n = ((a \bmod n)\cdot(b \bmod n)) \bmod n$$

if the intention is that the four $\bmod$'s denote the operation (notice that in this case, one should use $=$ instead of $\equiv$).

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