For example,
$16 \div 3 = (5, 1)$
where $16$ is the dividend, $3$ is the divisor, $5$ is the quotient, $1$ is the remainder.
But what about $2$ ?
Here $2 = 3 - 1$.
Is there an official name for this kind of remainder?
e.g. co-remainder? (or something like that)
Thanks.
I don't know of a tidy name, but I kind of know why you might want to use it.
In modular arithmetic, we bunch numbers together based on their difference being a multiple of a fixed number (in this case, $3$). Then $3-1=2$ is in the same bunch ("congruence class modulo $3$") as $-1$. And $2+1=3$ is in the same class as $0$. So this number $2$ is related to the "negative" or "additive inverse" or the original number $1$ and/or the congruence class of $1$. We sometimes want to move from thinking about to whole class back to the one number in the class that could be a remainder (in this case, $0$, $1$, or $2$), called the "residue".
So a long name for the idea might be like "the residue of the negative class (modulo $3$)".
Considering the divisor $d$ to be the modulus, and the remainder $r$ to be the residue, the difference $a=d-r$ is by definition the additive inverse of the residue, i.e. the number which when added to $r$ gives a sum which falls in the residue class $0$, viz: $r+a\equiv 0 \bmod d$
With the exception of the case when the remainder is zero (or there's no remainder, depending on your philosophical view) and provided the dividend and divisor are positive, then your "divisor minus remainder" is equal to the padding defined in the international standard ISO/IEC 10967, Language independent arithmetic (LIA). Part 2 of that standard defines the padding of $x \div y$ as $pad(x,y) = (\lceil x/y \rceil \times y)−x$ where $\lceil...\rceil$ is the "ceiling" function, i.e. if it's not a whole number then round up. So in your example, $16 / 3 = 5.333...$ so rounds up to $\lceil 16 / 3 \rceil = 6$, hence we need $pad(16,3)=2$ to get $16 = 6 \times 3 - 2$. On the other hand, the usual* definition of remainder would be $3 - 2 = 1$ because we prefer to write $16 = 5 \times 3 + 1$ to keep the remainder non-negative.
But in the case where the division can be performed exactly, the remainder and padding are both zero, rather than the padding being equal to the divisor (minus zero). If you are curious about the choice of terminology, and why we say the "padding" is $2$ rather than $-2$ in your example, the explanatory notes say the padding function
returns the negative of the remainder after division and ceiling. The reason for this is twofold: 1) for unsigned integer datatypes the remainder is $\le 0$, and would thus often not be representable unless negated, and 2) it is intuitively easier to think of the places left in the last unfilled group of equi-sized and packed groups" as a positive entity, a padding.
In a maths education context, I've also seen this padding called the deficit. See e.g. Liljedahl, P., Chernoff, E., & Zazkis, R. (2007). Interweaving mathematics and pedagogy in task design: A tale of one task. Journal of Mathematics Teacher Education, 10, 239-249.
More detail and some alternative terminology suggestions are available in this answer to the Math SE question Name for integer "quotient" rounded up (ceiling) instead of down (floor), and its negative or complementary "remainder".
$(*)$ It's usual in mathematics for the remainder to be defined to be non-negative. In computer science, there are several international standards, including ISO/IEC 10967 and ISO/IEC 60559 (which duplicates IEEE 754) in which the remainder is defined as what's left after rounding the quotient to the nearest whole number, and so the remainder can take both negative and positive values. For example, with $17 \div 3$ the closest integer quotient is $round(17/3) = round(5.666...) = 6$ and since $17 = 6 \times 3 + (-1)$ the so-called "IEEE remainder" is actually $-1$. On the other hand, the IEEE remainder for $16 \div 3$ is $+1$ since $round(16/3)=round(5.333...)=5$ and $16 = 5 \times 3 + 1$. The purported advantage of the IEEE remainder is that it gives the remainder, positive or negative, that is smallest (or if there's a tie, joint-smallest) in magnitude.
