## Task 3 - 220513

Rolf, a member of the District Club of Young Mathematicians, writes the following three equations for his classmates:

$$
\begin{aligned}
B \cdot J \cdot M & =135 \\
M+A+T+H+E & =32 \\
(H+E+I):(T-E-R) & =3
\end{aligned}
$$

He demands that each of the letters $A, B, E, H, I, J, M, R, T$ be replaced by one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 so that all three equations are true. The same letters should be replaced by the same numbers, and different letters by different numbers.

a) Anke responds: "I can already tell from the first equation which three numbers to use for B, J, and M. I just don't know their order yet."

Which three numbers are these?

b) Bertolt says: "Then, from the second equation, one can determine which number M represents." How could Bertolt have excluded the other two numbers mentioned by Anke?

c) After further trial and error, the classmates find a complete solution. What could be an example of such a solution?
a) The prime factorization of 135 is $135=3 \cdot 3 \cdot 3 \cdot 5$. Since $3 \cdot 5$ (and even more so $3 \cdot 3 \cdot 5$ or $3 \cdot 3 \cdot 3 \cdot 5$) is greater than any of the numbers $1,2, \ldots, 9$, the prime factor 5 must be one of the numbers for $B, J$, $M$. From the remaining three factors of 3, the other two numbers for $B, J, M$ can only be formed as 3 and $3 \cdot 3=9$.

Thus, 3, 5, and 9 are the three numbers for $B, J, M$.

b) For $A, T, H, E$, only four of the numbers 1, 2, 4, 6, 7, 8 are then in question.

If $M=3$ or $M=5$, then due to the second equation, the numbers for $A, T, H, E$ would have to sum to 29 or 27. This is not possible, as even the sum of the four largest of the numbers 1, $2,4,6,7,8$ is only 25. Therefore, $M=9$ must be true.

c) One solution is, for example:

$$
A=8, \quad B=3, \quad E=2, \quad H=6, \quad I=4, \quad J=5, \quad M=9, \quad R=1, \quad T=7
$$

because the equations $3 \cdot 5 \cdot 9=135, 9+8+7+6+2=32, (6+2+4):(7-2-1)=3$ are true. There is exactly one other solution. It arises from the one mentioned by swapping $B$ and $J$.