need to understand this soln?

Question

For a positive integer n, let Pn denote the product of the digits of n, and Sn denote the sum of the digits of n. The number of integers between 10 and 1000 for which Pn + Sn = n is

(a)81

(b)16

(c)18

(d)9

books soln :
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Let the number be xy. Then xy + x + y = 10x + y -> xy = 9x -> y = 9. The values that will satisfy the given conditions are: 19, 29, 39, 49, 59, 69, 79, 89, 99. Hence there are 9 such values.


the soln looks fine.

But see, book has taken only 2 digit number .

what about 3 and 4 digit and more digit numbers ?
it seems to me the soln given is not fair.

I need to understand why the more digit numbers are ignored.

Answer 1

Don't get mad about it. Try it and find out for yourself. If a three-digit number is xyz, then
xyz+x+y+z=100x+10y+z. Then
xyz= 99x+9y. However, because we "lost" the connection with z, this is generally not true, so there are no three-digit solutions.