Prove $∃x(∀y(\text{$5x+4y$ is even}))$

Question

I’m trying to solve this question but I’m not sure if I’m going correctly.

I know it’s false, so I tried to prove it defining which is odd (2k + 1) and which is even (2k) . Then I started working on the numbers, considering x and y as even, which would result in an odd result. Am I on the right the path?

Answer 1

The outermost quantifier is $\exists$, so you're supposed to come up with one $x$ that satisfies $\forall y(2\mid5x+4y)$. We can take $x=0$. I will let you continue from here.

Answer 2

Write the expression as $4(x+y)+x$, so we can see that it is even precisely when $x$ is even. Thus the statement becomes "there exists $x$ such that for all $y$, $x$ is even", which is obviously true: just take any even $x$.