## Task 1 - 160611

$$
\begin{array}{ccccc}
\mathrm{AAA} & \cdot & \mathrm{A} & =\mathrm{BBB} \\
+ & & & \\
\mathrm{CCC} & \cdot & \mathrm{E} & = & \mathrm{DDD} \\
\hline \mathrm{FFF} & : & \mathrm{F} & =\mathrm{GGG}
\end{array}
$$

In this scheme, digits $(0,1,2,3,4,5,6,7,8,9)$ are to be entered for the letters such that the same digits stand for the same letters and different digits for different letters, and that all five specified arithmetic operations are correctly calculated.

Determine all possible such entries!
If there is such an entry, it follows from the 3rd line that $G=1$. Furthermore, from the 1st line, $B$ is the square of $A (\neq B)$, so $B=4$ or $B=9$, and therefore $D=3$ or $D=8$.

On the other hand, $D$ (according to the 2nd line) is the product of two digits $C, E$ that are different from $D$, so $D$ cannot be a prime number.

Therefore, the only possibility left is $D=8, B=9, A=3$.

Since $8=2 \cdot 4$ is the only factorization of 8 into two factors different from 8 (up to the order), it follows that either $C=2, E=4$ or $C=4, E=2$. In the first case, $F=5$, and in the second case, $F=7$. Therefore, only the entries

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0212.jpg?height=202&width=910&top_left_y=1261&top_left_x=576)

can satisfy all the conditions of the problem. They do satisfy these conditions; because the digits $3,9,2,8,4,5,1$, or 3, 9, 4, 8, 2, 7, 1, used for $A, B, C, D, E, F, G$ are all different from each other, and the given arithmetic problems are correctly calculated.