Which is the smallest number that, when multiplied by 128, results in a number ending with 10 zeros?
I. solution: $128128=128 \cdot 1001=2^{7} \cdot 7 \cdot 11 \cdot 13$.

Since the factors of a number ending in ten zeros must include the factors $\left(10=2^{10} \cdot 5^{10}\right)$, the given number must be multiplied by at least $2^{3} \cdot 5^{10}$ to obtain a number ending in ten zeros. Therefore, $2^{3} \cdot 5^{10}=78125000$ is the sought number.

Miklós Csapody (Bp. VIII. Piarist g. I. o. t.)

II. solution: The prime factorization of the given number is $2^{7} \cdot 7 \cdot 11 \cdot 13$, and the smallest number ending in 10 zeros is $2^{10} \cdot 5^{10}$. The least common multiple of these two numbers is $2^{10} \cdot 5^{10} \cdot 7 \cdot 11 \cdot 13$, and thus the sought number is $\frac{2^{10} \cdot 5^{10} \cdot 7 \cdot 11 \cdot 13}{2^{7} \cdot 7 \cdot 11 \cdot 13}=2^{3} \cdot 5^{10}=78125000$.