Data Sufficiency

Hi

I have problems with three questions. This is the third of three related posts. I would appreciate your help in any! Thanks.

I'm having some difficulties understanding Q116 of the Official Guide OG Quant Review 1st edition in Data Sufficiency (page 193).
Question goes something like this:
116. If x is a positive integer, is sqrt(x) an integer?
(i) sqrt(4x) is an integer;
(ii) sqrt(3x) is not an integer.
The correct answer is A [statement (i) alone is sufficient]

statement (ii) is not a problem. I get why it is insufficient.
statement (i), however, is said to be sufficient because "Since sqrt(4x) is an integer, it follows that 4x must be the square of an integer. 4 is the square of 2. For 4x to be the square of an integer as well, x must also be the squre of an integer. therefore sqrt(x) must be an integer".

I worked it out differently, along these lines:
sqrt(4x) = integer = 2*sqrt(x)
Thus, sqrt(x) does NOT have to be an integer, as for example sqrt could be a 0.5 fraction (ie. an odd number divided by 2), which when multipled by 2 is an integer, but which in itself is not necessarily an integer.

My reasoning and the OG reasoning are obviously contradictory, but I don't understand why my reasoning is wrong. Would someone mind clarifying?

Thanks!

d problem states that x is an integer...you cant consider 0.5, which is non integer..

rights. thanks!