Question

N is a positive integer where 10 < N < 501. Let P and S denote the product of the digits of N and the sum of the digits of N respectively. The number of integers in the given range for which P + S = N is

• A. 81
• B. 49
• C. 29
• D. 16
• E. 9

Answer 1

Answer: (E)

To solve this problem, we need to find the number of integers \( N \) such that \( 10 < N < 501 \), where the sum of the product of the digits \( (P) \) and the sum of the digits \( (S) \) of \( N \) equals \( N \). In other words, \( P + S = N \). 

The number \( N \) can be categorized into different ranges: two-digit numbers (\(10 < N \leq 99\)), three-digit numbers (\(100 \leq N \leq 499\)), and specifically, a separate consideration for \( N = 500 \).

Step-by-step Solution:

1. For two-digit numbers (i.e., \( N = 10a + b \)), where \( 0 < a \leq 9 \) and \( 0 \leq b \leq 9 \):
   • The product of the digits is \( P = a \times b \).
   • The sum of the digits is \( S = a + b \).
   • The equation becomes \( a \times b + a + b = 10a + b \).
   • Simplifying, we get \( ab + a = 10a \rightarrow ab = 9a \rightarrow b = 9 \) (since \( a \neq 0 \)).
   • Hence, \( N = 10a + 9 \). The possible values of \( a \) are \( 1 \) to \( 9 \), giving us \( N = 19, 29, 39, 49, 59, 69, 79, 89, 99 \).
2. For three-digit numbers (i.e., \( N = 100a + 10b + c \)), where \( 1 \leq a \leq 4 \), \( 0 \leq b \leq 9 \), \( 0 \leq c \leq 9 \):
   • The product of the digits is \( P = a \times b \times c \).
   • The sum of the digits is \( S = a + b + c \).
   • The equation becomes \( a \times b \times c + a + b + c = 100a + 10b + c \).
   • This relation becomes more complex to solve analytically without assumptions, but upon checking specific values, no additional solutions in this range satisfy the equation.
3. Consider \( N = 500 \):
   • The product of the digits is \( P = 5 \times 0 \times 0 = 0 \).
   • The sum of the digits is \( S = 5 + 0 + 0 = 5 \).
   • The equation \( P + S = 0 + 5 = 5 \neq 500 \).
   • Thus, \( N = 500 \) does not satisfy the condition.

After evaluating all potential candidates, the only numbers that satisfy \( P + S = N \) are those found in the two-digit range. Therefore, there are nine such integers in the range \( 19, 29, 39, 49, 59, 69, 79, 89, 99 \), as verified by our solution steps.

The correct answer is 9.