Question 10
A special calculator displays the number $1$ on its screen when it is switched on and has only two buttons: @ and #. The @ and # buttons work differently:
| Button | Effect |
|---|---|
| @ | Multiply by $5$, then plus $2$ |
| # | Multiply by $7$, then minus $5$ |
Original button table:
For example, if you press @ first, then the calculator displays $1\times5+2=7$. Pressing @ again, it displays $7\times5+2=37$. Next, pressing # will then change the display to $37\times7-5=254$.
How many different three-digit numbers can be displayed by pressing @ and # buttons? (Just pressing one button many times is allowed.)
| Button | Effect |
|---|---|
| @ | Multiply by $5$, then plus $2$ |
| # | Multiply by $7$, then minus $5$ |
Original button table:
For example, if you press @ first, then the calculator displays $1\times5+2=7$. Pressing @ again, it displays $7\times5+2=37$. Next, pressing # will then change the display to $37\times7-5=254$.
How many different three-digit numbers can be displayed by pressing @ and # buttons? (Just pressing one button many times is allowed.)