# Why do we say -$11\div 3$ is $-4$ with remainder $1$, instead of $-3$ with remainder $-2$?

In the book Discrete Mathematics and Its Applications, 8e, Kenneth Rosen the quotient and remainder when $$-11$$ is divided by $$3$$ are specified as $$-4$$ and $$1$$ respectively. I would appreciate some help in understanding how we got there. It doesn't gel well with what I was taught in my elementary school.

Specifically, since $$-11=(-3)\times 3+(-2)$$, why do we not say that the quotient is $$-3$$ and the remainder is $$-2$$?

• $-11= ((-4) \times 3 )+1$ Commented Oct 20, 2021 at 6:59
• @MauroALLEGRANZA I want to know why not ((−3)×3)-2 Commented Oct 20, 2021 at 7:01
• You can: see Remainder: Examples Commented Oct 20, 2021 at 8:28
• In a more advanced setting one might treat "remainders" of $+1$ and $-2$ modulo $3$ as equal (literally, elements of the same equivalence class of residues). But in doing basic arithmetic one must adopt a convention to get a single result, and the simplest convention is to get the remainder $r$ on dividing by $b$ such that $0\le r \t b$. So following that convention (as many authors will), we chose the quotient that gives that nonnegative remainder. Something is wrong in your subject line BTW, as $11 \div 3$ should be quotient $3$ with remainder $2$. Apparently a minus sign was omitted? Commented Oct 21, 2021 at 0:13

What were you taught in elementary school that goes against it? Almost everywhere, this division rule is formally introduced with the name of Euclid's Division as follows:

Given two integers $$a$$ and $$b$$, with $$b ≠ 0$$, there exist unique integers $$q$$ and $$r$$ such that

$$a = bq + r$$ and

$$0 ≤ r < |b|$$, $$q = \lfloor\frac{a}{b}\rfloor$$

The above statement is taken from this Wikipedia article.

• a = -11
• b = 3
• q = -4 = floor(a/b)
• r = 1

I guess you hadn't noticed these constraints on the parameters of the equation. I hope this answers your question.

Dividend$$=$$Divisor$$\times$$Quotient +Remainder

Dividend $$=-11$$, Divisor$$=3$$, Quotient $$=-4$$, Remainder $$=1$$

$$-11=(3)(-4)+1$$ Edit: When you divide a number by another number, we get two different remainder values one of them being negative. In your case, remainder can be $$1$$ as well as $$-2$$(as $$1-3=-2$$)

• I want to understand why not -11=(3)(−3) - 2 Commented Oct 20, 2021 at 7:04
• Do you know concept of negative remainder$?$
– user982341
Commented Oct 20, 2021 at 7:06