Question

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?

• A. 0.1
• B. 0.11
• C. 0.16
• D. 0.22
• E. None of the above

Hint:

For digit-based problems, express the number as \(10a+b\) (two-digit form) to convert conditions into equations in \(a\) and \(b\). Often, the problem reduces to one simple fixed digit constraint, as here where the unit digit must be \(9\).

Answer 1

Answer: (E)

Step 1: Let the two-digit number be \(10a+b\), where \(a\) is the tens digit (\(1\leq a \leq 9\)) and \(b\) is the units digit (\(0\leq b \leq 9\)).
Step 2: The condition for "interesting" is \[ a+b+ab = 10a+b. \] Simplify: \[ ab + a + b = 10a + b ⇒ ab + a = 10a ⇒ ab = 9a. \] Step 3: If \(a=0\), not possible since the number is two-digit. For \(a \neq 0\), divide both sides by \(a\): \[ b = 9. \] So all interesting numbers end with digit \(9\). Step 4: Valid numbers are \(19, 29, 39, \dots, 99\). That is, 9 interesting numbers. Step 5: Total numbers between 10 and 100 inclusive: \(91\) (from 10 to 100). Thus fraction \[ \frac{9}{91}\approx 0.0989 \approx 0.099. \] This is approximately 0.099, which is closest to 0.1, but since the exact options provided do not match, the correct choice is \(\boxed{\text{None of the above}}\).