Question 7
A three-digit number is said to be interesting if it satisfies both the conditions below:
- All digits of this number are non-zero.
- After removing any one of the digits and adding the removed digit to the remaining two-digit number, the resulting number is always divisible by $8$.
For example, $195$ is interesting: removing $1$ gives $1+95=96$; removing $9$ gives $15+9=24$; removing $5$ gives $19+5=24$. All are divisible by $8$. How many interesting three-digit numbers are there in total?
- All digits of this number are non-zero.
- After removing any one of the digits and adding the removed digit to the remaining two-digit number, the resulting number is always divisible by $8$.
For example, $195$ is interesting: removing $1$ gives $1+95=96$; removing $9$ gives $15+9=24$; removing $5$ gives $19+5=24$. All are divisible by $8$. How many interesting three-digit numbers are there in total?