What do the Periods and Commas mean in this Mathematical Definition of the Modulo function? I have come across the following notation on a cryptography lecture slide that supposedly defines the modulo function:

$\forall a, n. \exists q, r. a = q \times n + r \text{ where } 0 \leq r < n$

Whilst I have a general idea of what it's doing, I am struggling to understand a few parts of the notation, which I have highlighted in red below:

$\forall a\color{red}{,} n\color{red}{.} \exists q\color{red}{,} r\color{red}{.} a = q \times n + r \text{ where } 0 \leq r < n$

What is the meaning of the commas and dots in this definition? It seems to me the dots are separating statements, so it reads:

For all $a$ and $n$ there exists a $q$ and $a$ for which $a = q \times n + r$ where $0 \leq r < n$.

Is that accurate?
 A: Your reading is the intended one. The commas here are being used to avoid repeating quantifier signs: $\forall a, n \cdots$ is being used as a shorthand for $\forall a\forall n \cdots$. The dot is being used to separate the quantifiers from the predicate part of the quantification. $\exists q, r. a = q\times n + r$ means $\exists q \exists r(a = q \times n + r)$. These abbreviations are fairly standard. The notation involving the dot is perhaps more common in computer science than in mathematical logic, but it is consistent with the standard notation in the $\lambda$-calculus.
The combination of symbolism and natural language using the word "where" should be avoided. If you want to be formal write "$\land$" not "where".
A: You are correct in your parsing of the symbols. And @FShrike's comment that there is sloppiness in the punctuation is accurate. The quoted
"∀,.∃,.=×+ where 0≤<"
would have better been written symbolically (if one insists) as
"$\forall a,n, \exists q,r$  such that $a=q\cdot n+r$, where $0\le r<n$"
(And, as HansLundmark commented, it ought not be $r\cdot a=...$, but something else... let's try again! :)  I myself was fooled by the punctuation, and was hasty, ...
That is, in this year, in LaTeX, a \cdot is a good multiplication symbol...
And, again, in words, as you already surmised, "for all $a$ and $n$, there are $q$ and $r$ such that ..."
(So, in words, $r$ is the remainder of $a$ after dividing-with-remainder by $n$, and $q$ is the (integer-) quotient.)
