## Task B-1.4.

Determine the smallest natural number $n$ for which the number $2565 \cdot n$ is a square of some natural number.
## Solution.

If we factorize the number 2565 into prime factors, we get $2565=3^{3} \cdot 5 \cdot 19$.

The number $2565 \cdot n$ is a square of some number if and only if each prime factor in its prime factorization appears an even number of times. Notice that in the obtained factorization, each prime factor appears an odd number of times.

Therefore, the smallest natural number that, when multiplied by the number 2565, gives a perfect square is obtained by selecting each of the prime factors exactly once.

Thus, the desired number $n$ is $n=3 \cdot 5 \cdot 19=285$.