Discussion

Explanation:

n can be a 2-digit or a 3-digit number.

Case 1: Let n be a 2 digit number.
Let n = 10x + y, where x and y are non-negative integers,
P_n = xy and S_n = x + y
Now, P_n + S_n = n
∴ xy + x + y = 10x + y
⇒ xy = 9x or y = 9
There are 9 two-digit numbers (19, 29, 29, … ,99) for which y = 9

Case (2): Let n be a 3-digit number.
Let n = 100x + 10y + z, where x, y and z are non-negative integers, 
P_n = xyz and S_n = x + y + z
Now, P_n + S_n = n
∴ xyz + x + y + z = 100x + 10y + z
⇒ xyz = 99x + 9y
⇒ z = 99/y + 9/x
From the above expression, 0 < x, y ≤ 9
But, we cannot find any value of x and y, for which z is a single-digit number. z will be minimum when x and y are both 9, but even then its value is 12.
∴ There are no 3-digit numbers which satisfy P_n + S_n = n

Hence, option (d).

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