## 5. Middle Number

Arrange in ascending order all three-digit numbers less than 550 whose hundreds digit is equal to the product of the other two digits. Among these arranged numbers, which number is in the middle?

## Result: $\quad 331$
## Solution.

Let the hundreds digit be 1. Since $1=1 \cdot 1$, the remaining digits are 1 and 1. The desired number is 111.

Let the hundreds digit be 2. Since $2=1 \cdot 2=2 \cdot 1$, the remaining digits are 1 and 2.

The desired numbers are 212 and 221.

Let the hundreds digit be 3. Since $3=1 \cdot 3=3 \cdot 1$, the remaining digits are 1 and 3.

The desired numbers are 313 and 331.

Let the hundreds digit be 4. Since $4=1 \cdot 4=2 \cdot 2=4 \cdot 1$, the remaining digits can be 1 and 4 or 2 and 2. The desired numbers are 414, 441, and 422.

Let the hundreds digit be 5. Since $5=1 \cdot 5=5 \cdot 1$, the remaining digits are 1 and 5.

The desired number is 515.

The number 551 is not a solution because we are looking for numbers less than 550.

The numbers with the given property, arranged in ascending order from the smallest to the largest, are:

$111, 212, 221, 313, 331, 414, 422, 441, 515$.

The middle number in the sequence is 331.