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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:           
  1. 9
  2. 16
  3. 18
  4. 81


Solution

The correct option is A 9
Let n be a two digit number.

n=10a+bPn=ab,Sn=a+b

then, ab+a+b=10a+b

ab=9a b=9

There are 9 such numbers 19,29,39,....,99.

Then let n be a three digit number.

n=100a+10b+cPn=abc,Sn=a+b+c

Then abc+a+b+c=100a+10b+c

abc=99a+9b

bc=99+9ba

But the maximum value for bc=81.

And RHS is more than 99. Hence, no such number is possible.

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