How many two digit numbers are there such that when multiplied by 2, 3, 4, 5, 6, 7, 8 or 9 don't change their sum of digits?

For example $18$ has a sum of digits equal to $1+8=9$, and when multiplied by any of those given numbers the resulting numbers sum of digits is still $9$.

I've realised that every number which has the sum of its digits equal to $9$ has this property that no matter what number you multiply it by you always preserve its sum of digits, but I don't know why only these numbers have this property