Proving a mathematical statement

[Temp0]

Homework Statement

There exists a number m, which is an element of the positive integers, that for all positive integers n, n+m can be divided by 3. Prove whether this statement is true or false.

Homework Equations

The Attempt at a Solution

I ran into a similar question earlier on, which just had the initial part reversed (as in, for all positive integers n there is a positive integer m so that n+m is divisible by 3). I proved that statement by letting m = 2n, and then 3n / 3 = n, which is a positive integer, proving that n+m is divisible. However, I don't understand why reversing the initial condition suddenly makes the entire statement false. Can I not do the same m = 2n idea to prove this statement? Could anyone explain why this is? Thank you in advance.

 

[mfb]

m cannot depend on n here.
If such an m would exist, you would have to be able to say "m=1245" for example.

 

[Temp0]

Ohh, so you're saying that m would be like a constant value, whereas in the other case it could be a variable? How exactly does the order play into this?

 

[mfb]

The order is in the statement.

"There is a number m [such] that for all integers n, ..." => fixed m, and then for all integers n something has to be true.
"For every n there is an integer m" => m can depend on n.

 

[Temp0]

Ohh, I see, thank you so much for your help.