The quotient is the result of a division, so \(a/b\) is called the quotient of \(a\) by \(b\).
Sometimes, one is interested in the number of times one thing can be divided ‘exactly’ into another, giving a quotient and a remainder.
For example, when working in the integers, \(13\) divided by \(3\) has a quotient of \(4\) and a remainder of \(1\).
As another example, \(x^4+x^2-x\) divided by \(x^2+3\) has a quotient of \(x^2-2\) and a remainder of \(-x+6\).
If \(a\) divided by \(b\) has a quotient of \(q\) and a remainder of \(r\), then \(a=bq+r\). Normally, one requires \(r\) to be ‘smaller’ than \(b\) in some sense. When dividing integers, we require \(0\le r<|b|\). When dividing polynomials, we require \(r=0\) or the degree of \(r\) to be less than the degree of \(b\).
The remainder of a division is \(0\) if and only if \(b\) is a factor of \(a\).