What is meant by quotient and remainder

Answer 1

Some examples (because I'm not that good at explaining):

4 / 2 = 2
quotient: 2 (number of times it can be divided without overshooting)
remainder: 0 (it's a whole number)

10 / 3 = 3.33...
quotient: 3
remainder: 1 (the leftover of the "undershot" division)

3 / 5 = 0.6
quotient: 0 (divisor is greater than dividend)
remainder: 5

In Python there are operators which allow you to get the quotient and remainder of divisions:

"//" quotient operator
>>> 12 // 5
2

"%" remainder/modulo operator
>>> 12 % 5
3

There are lots of good examples in the internet (with pictures) it would definitely help to do more research to clarify it :)

Answer 2

So quotient remainder theorem is a theorem from modular arithmetics.

The quotient remainder theorem says:

Given any integer A, and a positive integer B, there exist unique integers Q and R such that.

In other words you have

A= B * Q + R where 0 ≤ R < B

You can use this for make somes proves in modular arithmetic. Some direct
consequence is that you can have with this propertie is: if you can write a number
using

A= B * Q + R where 0 ≤ R < B

you can write in this form other form bellow too.
A mod B = R