You met a magician on a train, and after a little chat, he took out \(3\) squares of different sizes, as shown below. (The figure may not be drawn to scale.)
Magician: "These squares' side lengths are distinct digits (from \(1\) to \(9\) inclusive) whose greatest common factor is \(1\)."
With a flip of his hand, he transformed the squares into one whole rectangle.
Magician: "Now this rectangle's area is the same as the combined area of those \(3\) squares. The width of the rectangle is equal to the sum of the \(3\) squares’ side lengths, while the height of the rectangle is another distinct digit."
You: "How amazing! Can you tell me that height then?"
Magician: "No. Even if you know it, you still can’t work out the area of the rectangle."
You: "Can you at least tell me just one side length of the squares then?"
Magician: "No. Even if you know just any one square's length, you still can’t work out the area of the rectangle."
You: "Thanks! Now I know the area of the rectangle."
The magician became baffled after he had been advertently tricked to slip out a big clue.
What is the area of the rectangle?
Inspired by Digitalize This.