Why should the sum of three numbers be even?

Question

It says in "104 Number Theory Problems" by Titu Andreescu, Dorin Andrica, Zuming Feng (October 25, 2006) in the solution to this problem

Example 1.6. Find all positive integers n for which 3n − 4, 4n − 5, and 5n − 3 are all prime numbers.

Solution: The sum of the three numbers is an even number, so at least one of them is even. The only even prime number is 2. Only 3n − 4 and 5n − 3 can be even. Solving the equations 3n − 4 = 2 and 5n − 3 = 2 yields n = 2 and n = 1, respectively. It is trivial to check that n = 2 does make all three given numbers prime.

My question is how do they figure out, just from the information given in the question, that the sum of these numbers be even?

Answer 1

Because $3n-4+4n-5+5n-3=12n-12$, which is even.

Answer 2

Just write the sum: $(3n-4)+(4n-5)+(5n-3)=n(3+4+5)-(3+4+5)=(n-1)(3+4+5)=12(n-1)$

and this is even