Júlio performs multiplications using only the squares of numbers. He has to calculate the product $85 \times 135$. To do this, he draws a rectangle of $85 \mathrm{~mm}$ by $135 \mathrm{~mm}$ and traces the largest possible square in that rectangle; he does the same in the remaining square and so on. In this way, he obtains eight squares. Draw the figure made by Júlio and write $85 \times 135$ as the sum of eight squares: $85 \times 135=85^{2}+\ldots$

## Solutions to List 1
The largest square in the rectangle of $85 \times 135$ is the one of $85 \times 85$. This leaves a rectangle of $50 \times 85$, where the largest square is $50 \times 50$. Continuing this way, we get:

$$
85 \times 135=85^{2}+50^{2}+35^{2}+15^{2}+15^{2}+5^{2}+5^{2}+5^{2}
$$

|  |  |  | $5^{2} 5^{2} 5^{2}$ |  |
| :--- | :--- | :--- | :--- | :--- |
|  | $35^{2}$ | $15^{2}$ |  |  |
|  |  |  | $15^{2}$ |  |
|  |  |  |  |  |
|  |  |  |  |  |
|  |  | $50^{2}$ |  |  |
|  |  |  |  |  |
|  |  |  |  |  |