## Task 3 - 140923

It is to determine the smallest positive integer whose cube is an integer multiple of 588.
The prime factorization of 588 is: $588=2^{2} \cdot 3 \cdot 7^{2}$. If $z^{3}$ is the cube of a positive integer $z$, then $z^{3}$ must contain each prime factor of $z$ at least three times.

The smallest multiple with each prime factor of the factorization of 588 appearing three times is $2^{3} \cdot 3^{3} \cdot 7^{3}$. The cube root of this is $2 \cdot 3 \cdot 7=42 . z=42$ is the desired number.