.
The sum of three numbers is 165. When the smallest number is multiplied by 7, the result is $n$. The value $n$ is obtained by subtracting 9 from the largest number. This number $n$ also results by adding 9 to the third number. What is the product of the three numbers?
------------------------------------------------------------------
You use terms "smallest number", "largest number" and "third number".
They are not the terms of one semantic row. Which is bad.
Nevertheless, I conclude the "third number" is the "medium/middle number".
If it is not so, then your formulation is simply provocation.
From the condition, this system of three equations for three unknowns arises
s + m + l = 165, (1)
7s = l - 9, (2)
l - 9 = m + 9. (3).
where s is the smallest, l is the largest and m is the medium ("the third") number.
Solve it, if you can.
If not, then confirm that the "third number" is the medium.
Or re-edit and re-post your quest.