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Interesting Numbers Property

XAT 2013 · Quantitative Ability & Data Interpretation · Hard · Number Theory

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A number is interesting if on adding the sum of the digits of the number and the product of the digits of the number, the result is equal to the number. What fraction of numbers between 10 and 100 (both 10 and 100 included) is interesting?

  1. A.

    0.1

  2. B.

    0.11

  3. C.

    0.16

  4. D.

    0.22

  5. E.

    None of the above

Answer

E

Explanation

Let the two-digit number be 10x+y10x + y. Condition: (x+y)+xy=10x+y  ⟹  xy=9x  ⟹  y=9(x + y) + xy = 10x + y \implies xy = 9x \implies y = 9. Thus, any two-digit number ending in 9 (19, 29, 39, ..., 99) is interesting. There are 9 such numbers. Total numbers from 10 to 100 inclusive = 91. Fraction = 9/91≈0.09899/91 \approx 0.0989. Since 0.0989 is not listed, the correct option is None of the above.

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