\section*{Problem 1 - 330921}

If a three-digit natural number is multiplied by 7, the resulting number ends with the digits ... 638.

What is the three-digit number?
}

The sought number is [abc].

The last digit of the result can only be 8 if it was multiplied by a \(c=4\). All products of 7 with any other single-digit number end with a different digit.

The tens place 3 contains a carry-over 2 from the product \(7 \cdot 4=28\). Therefore, \(b=3\). Similarly, we conclude that \(a=2\). The sought number is 234, and it is \(234 \cdot 7=1638\).