## Task 2 - 150612

$$
\begin{array}{ccccc}
a & \cdot & a & = & b \\
- & & & & - \\
c & \cdot & a & = & d \\
\hline e & \cdot & a & = & a
\end{array}
$$

In the illustrated cryptogram, instead of the letters, digits are to be entered into the boxes so that all five given tasks are correctly solved.

The same letters should represent the same digits, and different letters should represent different digits.

As a solution, it is not sufficient, as is usually the case with such "number puzzles," to simply state the sought numbers. It must be demonstrated that the given numbers satisfy all the stated requirements and that they are the only numbers that do so.
The five tasks contained in the cryptogram are:
(1) $a \cdot a=b$,
(2) $c \cdot a=d$,
(3) $e \cdot a=a$,
(4) $a-c=e$,
(5) $b-d=a$.

If five digits $a, b, c, d, e$ correctly solve these tasks and are all different from each other, then $a \neq 0$; for if $a=0$, then $b=0$ due to (1). From (3) it follows that $e=1$.

Furthermore, $c \neq 0$; for if $c=0$, then $d=0$ due to (2).

Given this and because $c \neq e$, it follows that $c \geq 2$. From (4), $a=c+e=3$. On the other hand, according to (1) and because $b$ is a single digit, $a \cdot a \leq 9$, so $a \leq 3$. Therefore, $a$ must be 3, and from (4), $c=a-e=2$.

From (1) and (2), we get $b=9, d=6$. Therefore, only the entry can satisfy the conditions of the problem statement.

These values satisfy the conditions; because the digits $3, 9, 2, 6, 1$ substituted for $a, b, c, d, e$ are all different, and all the tasks in the cryptogram are correctly solved with these digits. Thus, the given entry has the required properties.

$$
\begin{array}{cccc}
3 & \cdot & 3 & = \\
- & & \\
2 & \cdot & 3 & = \\
\hline 1 & \cdot & 3 & = \\
\hline
\end{array}
$$