Problem 8 · Multiples of their digit sum
A systematic count by digit sums (Harshad numbers).
Integer answer, at most 4 digitsHow many two-digit numbers are multiples of the sum of their two digits? For example, 42 is a multiple of $6 = 4 + 2$, but 14 is not a multiple of $5 = 1 + 4$.
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: write $n = 10a + b$ with digit sum $s = a + b$. The condition $s \mid n$ is equivalent to $s \mid n - s = 9a$: the digit sum just needs to divide $9a$.
Going through them in order (or grouping by digit sums), the numbers satisfying the condition are:
$$10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90.$$
(All two-digit multiples of $10$ and of $9$ appear, among others.) In total, $23$ numbers.
Answer: 23