Task 3. Represent the number 1000000 as a product of two numbers, in whose notation not a single zero appears.
Solution. The number 1000000 can be represented as

$$
1000000=10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10=(2 \cdot 5) \cdot(2 \cdot 5) \cdot(2 \cdot 5) \cdot(2 \cdot 5) \cdot(2 \cdot 5) \cdot(2 \cdot 5)
$$

If $x$ and $y$ are natural numbers such that $xy=1000000$ and the number $x$ does not contain the digit 0 in its representation, then 2 is a divisor of $x$, but 5 is not a divisor of $x$, or 2 is not a divisor of $x$, but 5 is a divisor of $x$. If 2 is a divisor of $x$ and 5 is a divisor of $x$, then 10 is a divisor of $x$ and the units digit of $x$ is zero. The same discussion applies to the number $y$. Therefore, the only possibilities are: $x=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2=64$ and $y=5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5=15625$, or $x=15625$ and $y=64$. In each case, the required representation is $1000000=15625 \cdot 64$.