GCD or LCM confusion

Question

Rosa is making a game board that is 16 inches by 24 inches. She wants to use square tiles.

What is the largest tile she can use and how many square tile will be on her board?

Need explanation on why GCF is applied not the LCM

Answer 1

The side length of the square needs to divide both 16 and 24 evenly. The largest such integer is (by definition) the greatest common factor.

Answer 2

If the side length of the tile is $d$ you need to fit an exact number of tiles in each direction. So you need $md=16$ and $nd=24$. So $d$ has to be a common factor of $16$ and $24$. If you want the largest possible $d$, then you need the highest/greatest common factor/divisor.

The biggest confusion comes because the terminology is not constant. Highest and greatest are both used, as are factor and divisor. They all mean the same thing.