123. Problem with prime numbers. The product of a certain three-digit number by a certain two-digit number has the form

| $p p p$ |
| ---: |
| $p p$ |
| $p p p p$ |
| $p p p p$ |
| $p p p p p$ |

Here the letter $p$ can represent any prime digit other than one (not necessarily the same one). Restore all the numbers and show that the solution to this problem is unique.
123. We need to find a three-digit and a one-digit number, the product of which is a four-digit number, and only the digits $2,3,5$ or 7 are allowed. There are only four possible solutions: $3 \cdot 775=2325, 5 \cdot 555=2775$, $5 \cdot 755=3775$ and $7 \cdot 325=2275$.

Since none of the three-digit numbers in these four products appear more than once, we conclude that the sought two-digit number consists of identical digits. Thus, the only solution is of the form

| 775 |
| ---: |
| 33 |
| 2325 |
| 2325 |
| 25575 |

[W. E. Baker, A. M. M., 43, 499 (October 1936).]