## Task 2 - 190712

Determine all four-digit natural numbers that have the property of being divisible by each of the numbers $2,3,4,5,6,7,8,9,10,12,14,15$!
A natural number is divisible by the given numbers if and only if it is divisible by their $\mathrm{lcm}$. Due to the prime factorization

$$
\begin{array}{cccc}
2=2 & 5=5 & 8=2^3 & 12=2^2 \cdot 3 \\
3=3 & 6=2 \cdot 3 & 9=3^2 & 14=2 \cdot 7 \\
4=2^2 & 7=7 & 10=2 \cdot 5 & 15=3 \cdot 5
\end{array}
$$

this $\mathrm{lcm}$ is the number $2^3 \cdot 3^2 \cdot 5 \cdot 7=2520$. All (non-zero) natural multiples of this number are: The numbers $1 \cdot 2520=2520, 2 \cdot 2520=5040, 3 \cdot 2520=7560$ as well as all numbers $n \cdot 2520$ with natural $n \geq 4$.

However, for each $n \geq 4$: Since $n \cdot 2520 \geq 4 \cdot 2520=10080$, the number $n \cdot 2520$ is not a four-digit number. Therefore, exactly the numbers 2520, 5040, and 7560 satisfy the conditions of the problem.