Prove that $\prod\limits_{k=1}^n(4-\tfrac{2}{k}) \in \mathbb{N}$.

Question

How to prove that $$\prod\limits_{k=1}^n\left(4-\dfrac{2}{k}\right) \in \mathbb{N}.\tag{1}$$

Moreover, that it is even number.

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Update: sos440 give me great hint on $(1)$.

And how about this one: $$\prod\limits_{k=1}^n\left(4+\dfrac{2}{k}\right) \in \mathbb{N}.\tag{2}$$

Answer 1

For example: $$\prod\limits_{k=1}^n\left(4+\dfrac{2}{k}\right) = 2^n \frac{1 \cdot 3 \cdot 5 \cdots 2n+1 }{1 \cdot 2 \cdots n} = 2^n \frac{(2n)! (2n+1)}{n! (n!)2^n}=(2n +1) {2n \choose n}$$

Answer 2

The first one is already given in the comments. The relationship between first one and second one can be seen by

$$ k_n=\prod_{k=1}^n(4-\frac{2}{k})=\frac{2.6.10.14.18\dots (4n-2)}{1.2.3.4.5\dots n} $$

$$ \prod_{k=1}^n(4+\frac{2}{k})=\frac{6.10.14.18\dots(4n-2)(4n+2)}{1.2.3.4.5\dots n}=k_n/2+4n+2 $$