Costello Operator

Let $p$ be the two-digit number and $q$ be the one-digit number, and let $a$ be the first digit of $p$ and $b$ be the last digit of $p$ . Then $p * q = qb + qa = q(a + b)$ = 28. So we’re looking for factors of 28 such that one is a single digit ( $q$ ), and the other is the sum of two single digits ( $a$ and $b$ ). The single digit factors of 28 are 1, 2, 4, and 7 and are therefore possible $q$ values.

If $q$ = 1, then $a + b$ = 28, which is too big to be expressed as the sum of two single digits.

If $q$ = 2, then $a + b$ = 14, which can be expressed as the sum of two single digits with 5 and 9, 6 and 8, and 7 and 7; for 5 cases : 59 * 2, 68 * 2, 77 * 2, 86 * 2, and 95 * 2.

If $q$ = 4, then $a + b$ = 7, which can be expressed as the sum of two single digits with 0 and 7, 1 and 6, 2 and 5, and 3 and 4; for 7 cases : 16 * 4, 25 * 4, 34 * 4, 43 * 4, 52 * 4, 61 * 4, and 70 * 4.

Finally, if $q$ = 7, then $a + b$ = 4, which can be expressed as the sum of two single digits with 0 and 4, 1 and 3, and 2 and 2; for 4 cases : 13 * 7, 22 * 7, 31 * 7, and 40 * 7.

There are therefore 5 + 7 + 4 = $\boxed{16}$ total ways a Costello operator can be applied to a two-digit number and a one-digit number to obtain an answer of 28.