4 Divides x Proofs of conjectures

Question

Hi there I'm working on a set of problems and I'm having some difficulty proving and disproving these examples. I know that #1 is essentially (There exists K where [x=4k]) I'm lost after that. I'm not sure about 2 or 3 I believe that 2 is false and 3 is true but I don't know how to prove or disprove formally. Any advice would be wonderful thank you!

1. Define what it means for a number to be divisible by 4, that is, the concept 4 | x.

Answer: There exists k where [x=4k]

2. Determine whether the following conjecture is true or false. If it is true, prove it, otherwise construct a counterexample to disprove it. Conjecture 1. Let x and y be arbitrary integers. If 4 | (x−y), then 4 | (3x+y).

3. Determine whether the following conjecture is true or false. If it is true, prove it, otherwise construct a counterexample to disprove it. Conjecture 2. For integers x and y, if 4 | x and 8 | y, then 8 | (x+y).

Answer 1

Here are some hints:

2. Try to write $(3x + y)$ as $(x-y)$ plus some term. What can you deduce?

3. Mess around with some examples. Try to find $x$ divisible by 4 and $y$ divisible by 8, but where $x + y$ is not divisible by 8.

Answer 2

You are wrong. 2 is true and 3 is false in reality.

2. $x-y=4k\implies 3x+y= 4x-4k = 4(x-k)$

3. $4\mid4$ and $8\mid8$ but $8\nmid 12$