Express all possible positive integer values of m, n in terms of a third variable, k. (Use modular arithmetic)

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- Sep 14th 2007, 05:29 AMDivideBy0possible values for simple equation

Express all possible positive integer values of m, n in terms of a third variable, k. (Use modular arithmetic) - Sep 14th 2007, 03:51 PMSoroban
Hello, DivideBy0!

We can solve this with "normal" algebra . . .

Quote:

Express all possible positive integer values of in terms of a third variable,

Solve for .**[1]**

Since is an integer, must be a multiple of 4.

. . That is: . for some integer

Substitue into [1]: .

And we have parametric equations for all solutions:

. . . for any integer

- Sep 15th 2007, 04:31 AMDivideBy0
Thanks, I have two question though

How would you express them if you could only have them as natural numbers, would you have 1 =< k =< 33?

and What are parametric equations? thanks again - Sep 15th 2007, 06:32 AMCaptainBlack
- Sep 15th 2007, 08:41 AMSoroban
Hello, DivideBy0!

Quote:

How would you express them if you could only have them as natural numbers?

Would you have: . ? . . . . Yes!

What are parametric equations?

A parameter is an "extra variable".

Instead of having as a function of ,

. . we can have: .

This opens the door to an entire*universe*of fascinating functions and graphs

. . . . . curves with loops, that intersect itself, etc.