Are the multiples of an even number always even?

Yes, assuming you're multiplying it by another integer. By definition, an even number is a multiple of two. When you multiply it be another round number, that even number becomes a factor of the result. That means any factors of that even number (one of them being two), will also be factors of the result, thus giving you another even number.

If however you multiply by a non-integer, you may very well end up with something that is not an even number. For example, 6 is an even number, but 6 × 1.5 will give you 9, which is not an even number.