# Problem №2

Find all two-digit numbers, the sum of the digits of which does not change when the number is multiplied by $2,3,4,5,6,7,8$ and 9.
Answer: $18,45,90$ and 99.

Solution: According to the condition, the sum of the digits of the number $a$ and the number $9 a$ is the same. Therefore, according to the divisibility rule for 9, the number $a$ is divisible by 9. Two-digit numbers divisible by 9 are as follows: $18,27,36,45,54,63,72,81,90$ 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.