# Thread: Range of a multivariable function

1. ## Range of a multivariable function

How do you find the range, algebraically, of a multivariable function?

I'm attempting to prove the following:

Prove that there do not exist integers m and n such that 12m+15n=1.

I know there's a different way to do it, but I want to show that 1 is not in the range of the multivariable function f(m,n)=12m+15n.

I guess I could simply say, "Obviously, 1 is not in the range of this function" but I want to show it algebraically for the proof.

Any help would be appreciated!

2. Originally Posted by divinelogos
How do you find the range, algebraically, of a multivariable function?

I'm attempting to prove the following:

Prove that there do not exist integers m and n such that 12m+15n=1.

I know there's a different way to do it, but I want to show that 1 is not in the range of the multivariable function f(m,n)=12m+15n.

I guess I could simply say, "Obviously, 1 is not in the range of this function" but I want to show it algebraically for the proof.

Any help would be appreciated!
Your approach is flawed because 1 is in the range of the function eg. m = 1/12 and n = 0. Your approach does not take into account the simple fact that the 'domain' is required to be the integers not real numbers.

(This is one reason why I have moved your question to number theory).

The required proof probably expects you to note that the gcd of 12 and 15 is not 1.

3. Originally Posted by mr fantastic
Your approach is flawed because 1 is in the range of the function. You can't treat it as a multi-variable function because the domain is integers not real numbers.

(This is one reason why I have moved your question to number theory).

The required proof probably expects you to note that the gcd of 12 and 15 is not 1.
Can I simply restrict the domain of the function to integers, and then show that 1 is not in the range?

4. Originally Posted by divinelogos
Can I simply restrict the domain of the function to integers, and then show that 1 is not in the range?
That is exactly what the question has asked you to do!!

5. Originally Posted by mr fantastic
That is exactly what the question has asked you to do!!
Right, but I want to show it algebraically as follows:

1. If there were integers m and n such that 12m+15n=1, then 1 would be in the range of the function "f(m,n)=1m+15n whose domain is integers"

2.(this is where I want to show algebraically that 1 is not in the range of that function^^).

3.Hence, there are not integers m and n such that 12m+15n=1.

6. Originally Posted by divinelogos
Right, but I want to show it algebraically as follows:

1. If there were integers m and n such that 12m+15n=1, then 1 would be in the range of the function "f(m,n)=1m+15n whose domain is integers"

2.(this is where I want to show algebraically that 1 is not in the range of that function^^).

3.Hence, there are not integers m and n such that 12m+15n=1.

What's the problem with simply saying "as 3 divides both of 12 and 15 it divides 12m + 15n , for all

integers m,n, and ..."??

Tonio

7. Originally Posted by tonio
What's the problem with simply saying "as 3 divides both of 12 and 15 it divides 12m + 15n , for all

integers m,n, and ..."??

Tonio
There's nothing wrong with it. I just wanted to see how to do it the other way, or if it was even possible

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