1. Find all four-digit numbers $\overline{a b c d}$, for which $\overline{a b c d}=20 \cdot \overline{a b}+16 \cdot \overline{c d}$.
1. In the equation from the assignment

$$
1000 a+100 b+10 c+d=20(10 a+b)+16(10 c+d)
$$

the unknown digits $a$ and $b$ have larger coefficients on the left side, while the digits $c$ and $d$ have larger coefficients on the right side. Therefore, we rearrange the equation to the form $800 a+80 b=150 c+15 d$, which after dividing by five and factoring out the smaller coefficients on both sides, can be rewritten as $16(10 a+b)=3(10 c+d)$. From the coprimality of the numbers 3 and 16, it follows that $10 c+d$ is a two-digit multiple of 16. However, this must be greater than 48, since $3 \cdot 48=144$, while $16(10 a+b) \geqq 160$ (the digit $a$ must be non-zero). Therefore, the possible values for $10 c+d$ are only the multiples of 16 equal to 64, 80, and 96, which determine the digits $c$ and $d$. Substituting into the equation $16(10a+ +b)=3(10 c+d)$, we get the values 12, 15, and 18 for the two-digit number $10 a+b$.

Answer: The numbers that satisfy the condition are 1264, 1580, and 1896.

Note: Instead of the four unknown digits $a, b, c, d$, we can use the notation of the equation from the assignment directly with the two two-digit numbers $x=\overline{a b}$ and $y=\overline{c d}$. The equation will then have the form $100 x+y=20 x+16 y$, which we can similarly rearrange to $80 x=15 y$ or $16 x=3 y$. Now, instead of the relation $16 \mid y$, we can use a similar consequence $3 \mid x$ and realize that from the estimate $y \leqq 99$ it follows that $16 x \leqq 3 \cdot 99=297$, from which $x \leqq 18$, which together with the estimate $x \geqq 10$ leads to the possible values $x \in\{12,15,18\}$. From the equation $16 x=3 y$ we then calculate $y=64$ for $x=12$, $y=80$ for $x=15$, and $y=96$ for $x=18$.

For a complete solution, award 6 points, a logically correct procedure with calculation errors (depending on their extent) up to 4 points. For an incomplete procedure, award 1 point for writing the given equation in expanded form (i.e., with powers of base 10), and 1 point for rearranging it to a product form with coprime factors 16 and 3.