[ Divisibility rules for 3 and 9 ] $[\quad$ Case enumeration $]$

Find all two-digit numbers whose digit sum remains unchanged when the number is multiplied by 2, 3, 4, 5, 6, 7, 8, and 9.

#
According to the condition, the sum of the digits of the number $a$ and the number $9a$ is the same. Therefore, according to the divisibility rule for 9, the number $a$ is divisible by 9. The two-digit numbers that are multiples of 9 are: 18, 27, 36, 45, 54, 63, 72, 81, 90, and 99. Among them, the numbers 27, $36, 54, 63, 72$, and 81 do not have the required property; this can be verified by multiplying them, respectively, by $7, 8, 7, 3, 4$, and 9. The remaining numbers have the required property.

## Answer

$18, 45, 90$ and 99.