How long should the side of a square be, which can be divided without residue into rectangles 7cm wide and 16cm long?

The search is set incorrectly: in the problem it is necessary to find the width of the second rectangle, of the two that make up a square! Otherwise, there is no logic.
Decision:
In a square, side (A) is 16 cm, as is the length of each side of the two rectangles.
The width of one of the rectangles complements the side of the square: (A) = 16 = 7 + X, whence X, i.e. the width of the second of the rectangles obtained from the division of the square is: 16 – 7 = 9 cm.
Answer: sides of the second rectangle: width – 9 cm, length – 16 cm.