**Prove that **the expression is an integer for all . According to the **Division algorithm**:

Given integers a and b, with there exist unique integers q and r such that

where

The integers q and r are respectively, the quotient and remainder in the division of a by b.

**Proof: **

The advantage of the Division Algorithm is that it allows us to prove assertions about all the integers by considering only a finite number of cases.

In this example, the division algorithm tells us that every integer a is of the form , or . Assume the first of these cases. Then,

which clearly is an integer. Similarly, if , then

and the given expression is an integer in this instance also.

Finally, for , we obtain which is an integer once more. Consequently, our result is established in all cases.

**Some more applications.**

We wish to focus our attention on the applications of the Division Algorithm, and not so much on the algorithm itself. As another illustration, note that with , the possible remainder are and . When , the integer a has the form and is called even; when , the integer a has the form and is called odd. Now, is either of the form or . **The point to made is that the square of an integer leaves the remainder 0 or 1 upon division by 4.**

We also can show the following: **The square of any odd integer is of the form **. For, by the Division Algorithm, any integer is representable as one of the four forms , , and . In this classification, only those integers of the forms and are odd. When the latter are squared, w find that

and similarly,

.

As examples, the square of the odd integer 7 is , and the square of 13 is .

So, once again, to recur an important point: the division algorithm allows us to prove assertions about all the integers by considering only a finite number of cases! 🙂 That’s a vast simplification/improvement !!

More later,

Nalin Pithwa