We have been working with the division algorithm since we started learning division.

Here are some definitions to cover initially:

Divides: an integer \(d\) divides \(a\) provided that \(a=dq\)
 (where \(q\) is an integer)
We can also say that \(a\) is a multiple of \(d\).

For example, does \(8\) divide \(32\)?

We will see that \(32=8\times4\).Thus, \(8\) divides \(32\) and \(32\) is a multiple of \(8\).


The Division Theorem:
For positive integers \(a\) and \(d\), there are unique integers q and r such that \(a=dq+r\) ,  where \(0\le r<d\).

This is essentially spells out what we did above, but it definitely clearly defines the constraints of the remainder. Lets do an example:

Example:

If we know that \(a=50\) and \(d=7\), then what is \(q\) and \(r\). Then we know that the equation would be \(50=7q+r\). Thus \(q=7\) and \(r=1\). But what if we used \(q=6\) and \(r=8\), or can't r be negative and then we would have many options? .......


No. Remember our definition includes the constraint on the remainder.