1. Find the smallest natural number, which when multiplied by 127008 gives a cube of a natural number. Explain your answer!
Solution. Since $127008=2^{5} \cdot 3^{4} \cdot 7^{2}$, it follows that the smallest natural number by which we need to multiply this number is the number $2 \cdot 3^{2} \cdot 7=126$.