Is it possible to multiply the number 101001000100001 by another integer so that the result has no digits equal to zero?

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Solution

Initially consider the number 101, which has a "hole" with only one zero. Multiplying by 11 eliminates the "hole," as shown in the diagram below of the multiplication algorithm:

|  | 1 | 0 | 1 |
| :---: | :---: | :---: | :---: |
| $\times$ | 1 | 1 |  |
|  | 1 | 0 | 1 |
| 1 | 0 | 1 |  |
| 1 | 1 | 1 | 1. |

Now consider the number 1001. If we wanted to do the same, that is, multiply it by another number to obtain a result without any digits equal to zero, we could simply choose the number 111. Again, to confirm this, consider the diagram below as the multiplication algorithm:

|  |  | 1 | 0 | 0 | 1 |
| :---: | :---: | :---: | :---: | :---: | :---: |
|  |  | $\times$ | 1 | 1 | 1 |
|  |  | 1 | 0 | 0 | 1 |
|  | 1 | 0 | 0 | 1 |  |
| 1 | 0 | 0 | 1 |  |  |
| 1 | 1 | 1 | 1 | 1 | 1. |

Multiplying an integer by a sequence of 1's using the multiplication algorithm is the same as writing this integer in several lines, giving a small shift to the left from one line to the next, and then adding them all together. Analyzing the initially given number, if we wanted to repeat the previous procedure only for the number 100001, it would suffice to multiply it by 11111. Indeed, multiplying by 11111 also serves to "cover" all the holes of zeros between the digits equal to 1, as shown in the diagram below:

|  |  |  |  | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $\times$ | 1 | 1 |  |  |  |  |  |  |  |  |  | 1 | 1 |  |  |  |  |  |
| 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1. |