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So I was practicing for a test by reviewing textbook questions and I ran into this problem. I understand the concept of quotient and remainder but how would you approach this if you are dealing with a negative value?

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    Find largest number smaller than -17 that is a multiple of 3. then substract and continue... – 2017-02-27

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If you're doing discrete mathematics / number theory, then we usually define the quotient and remainder as follows: The quotient and remainder when $a$ is divided by $b$ is $$ a=bq+r $$ where $q$ and $r$ are integers, $q$ is the quotient, and $r$ is the remainder. Moreover, $0\leq r

In your case, $a=-17$ and $b=3$. You need to find $q$ and $r$ that satisfy $$ -17=3q+r $$ where $0\leq r<3$, or that $r$ is $0$, $1$, or $2$. Moving the $3q$ to the LHS, we get $$ -17-3q=r. $$ Since $r$ is nonnegative, $q$ must be negative, so we try a few values and find that $q=-6$ works nicely. In fact, $$ -17-3(-6)=-17+18=1. $$ Therefore, the quotient is $-6$ and the remainder is $1$. The reason that this works is that $18$ is the first multiple of $3$ which, when added to $-17$ gives a nonnegative number.

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    What did u mean when you wrote q= -6 works nicely, like what do you mean we try a few values – 2017-02-27
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    Also what would constitute as a "nice value" if the example was different – 2017-02-27
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    You need a remainder of $0$, $1$, or $2$. So, you pick values for $q$, plug them in and see what you get for $r$. I'm suggesting a guess and check approach - there are formulas, but they might be more confusing. – 2017-02-27