# SP20:Lecture 13 prep

Here are the notes for last semester.

We'll start lecture 13 by fixing the broken proof that every number has a prime decomposition that we started in lecture 12.

Be sure you understand why we are stuck at the end of that lecture.

We'll also prove the existence and uniqueness of the quotient and remainder. Come to lecture knowing the following definitions and how they relate to your existing notion of quotient and remainder:

Definition: Quotient
We say is a quotient of over if for some with . We write (note that quot is a well defined function).
Definition: Remainder
We say is a remainder of over if for some and . We write (note that rem is a well defined function).

Time permitting, we may start discussion the base b representation of a number; for this we will need the following definitions:

Definition: Digit
numbers satisfying are called base b digits.
If are all natural numbers satisfying for all , then the base b interpretation of , written is given by