9.1. What different values can the digit U take in the puzzle U$\cdot$LAN + U$\cdot$DE = 2020? Justify your answer. (Identical digits are replaced by the same letters, different digits by different letters.)
Answer: two. $\mathrm{V}=2, \mathrm{y}=5$.

Solution: Factor out the common factor: У$\cdot$(ЛАН + ДЭ) $=2020$. Note that У, Л, and Э are not equal to 0.

Factorize the right-hand side: $2020=1 \cdot 2 \cdot 2 \cdot 5 \cdot 101$. Since У is a digit, consider all possible values: $\mathrm{V}=1,2,4,5$.

1) $У=1$. Then ЛАН + ДЭ $=2020$, but ЛАН + ДЭ $<999+99=1098<2020$. No solutions.
2) $\mathbf{y}=2$, ЛАН + ДЭ $=1010$.

There are 8 different solutions, for example, $2 \cdot 974+2 \cdot 36=2020$

3) $У=4$, ЛАН + ДЭ $=505$. Clearly, Л $\leq 5$.

Л $\neq 5$, because otherwise ЛАН $\geq 501$, but then ДЭ cannot be a two-digit number.

$Л \neq 4$, because $У=4$.

For Л $=3$, АН + ДЭ $=205$. But АН + ДЭ $\leq 99+99=198$ < 205. No solutions.

4) $\mathbf{y}=\mathbf{5}$, ЛАН + ДЭ $=404$.

There are 14 different solutions, for example, $5 \cdot 306+5 \cdot 98=2020$

# Criteria: 

7 points - correct answer and complete correct solution;

only one answer found and justified - 4 points;

only answer - 0 points.