Recently my colleague ask one mathematical question which is,
What is the quotient and remainder of $(-29)/7$?
and my answer was that quotient is $4$ and remainder is $-1$ and he told me I'm completely wrong. He said that the reminder will be $6$, and I'm really get stuck here how will the remainder be $6$.
EDIT:
Sorry for my great mistake, the question will be $(-29)/7$ and his result was quotient is $-5$ and remainder is $6$
You should ask him exactly what he means when he says remainder. It is common for us to mean the following:
For any two integers $a,b$, with $a \neq 0$, there exist unique integers $q,r$ such that $$ b = aq + r$$ where $0 \leq r < |a|$. Then we call $q$ the "quotient" and $r$ the "remainder." This fact is sometimes called the division algorithm, but it implicitly contains our definitions of quotient and remainder. In particular, we are demanding that the remainder be positive.
With $-29$ and $7$, we have that $$ -29 = 7\cdot(-5) + 6,$$ so that according to my definition of remainder we would get $6$.
As an aside, there is no reasonable way to get $-10$ as a remainder, regardless of definition.