## Task 1 - 210921

In the division problem $a: b=c$, $a, b, c$ are to be replaced by natural numbers so that a correctly calculated division problem results, where only the digits 1, 2, 3, 4, 5, and each exactly once, are to be used.

Determine all triples $(a ; b ; c)$ of natural numbers that meet these requirements!
Given that there are 5 digits available for three numbers, at least one must be a single-digit number. Since the dividend $a$ in a division problem (within the realm of natural numbers) is the largest value, $a$ must be at least a two-digit number. If both $b$ and $c$ were single-digit, their product $a$ would be at most $5 \cdot 4 = 20$, meaning not all five digits would be used. Therefore, $a$ must be a two-digit number, and exactly one of the two values $b$ or $c$ must also be a two-digit number, while the other is a single-digit number.

Since the digit 5 can only be used once, it cannot be used as a unit digit, as this would make the corresponding number divisible by 5, which would in turn require at least one of the other three numbers to also be divisible by 5. This is not possible due to the lack of available unit digits 0 or another 5.

Therefore, the digit 5 must be used as a tens digit. However, it cannot be the tens digit of the quotient or divisor, as this would make the value at least 51, while the largest possible value for the dividend, which must be larger, would be 43. Thus, the 5 must be the tens digit of $a$.

Neither $b$ nor $c$ can be 1, as this would make the other value equal to $a$, leading to the reuse of digits.

Since 53 is a prime number and $51 = 3 \cdot 17$ as the only non-trivial factorization uses the unavailable digit 7, these two possibilities for $a$ are excluded. This leaves two cases:

1. Case: $a = 52$. Then, because $52 = 2 \cdot 26 = 4 \cdot 13$, there are exactly two possibilities: $b = 4$ and $c = 13$ or $b = 13$ and $c = 4$. No other factorizations exist in this case.
2. Case: $a = 54$. Then, because $54 = 2 \cdot 27 = 3 \cdot 18 = 6 \cdot 9$, there are no non-trivial factorizations that meet the requirements of the problem.

Therefore, there are exactly two such triples, namely $(a, b, c) \in \{(52, 4, 13), (52, 13, 4)\}$.