Question

Let p ( n ) and s ( n ) denote the product and the sum, respectively, of the digits of the integer n . For example, p ( 23 ) = 6 and s ( 23 ) = 5 . Suppose N is a two-digit number such that n = p ( n ) + s ( n ) . What is the unit digit of n ?

Answer 1

Let's consider a two-digit number N where the sum of its digits is s(N) and the product of its digits is p(N). According to the given condition:

N = p(N) + s(N)

We know that the largest possible product of two single-digit numbers is 9, which occurs when both digits are 9. Therefore, p(N) ≤ 9.

The largest possible sum of two single-digit numbers is 18, which occurs when both digits are 9. Therefore, s(N) ≤ 18.

Now, let's find the unit digit of N. Since we are looking for the unit digit, we need to consider the possible values of p(N) and s(N) that result in a unit digit for N.

1. If p(N) = 9 (the maximum value for the product of two digits) and s(N) = 9 (the maximum value for the sum of two digits), then N = 9 + 9 = 18. In this case, the unit digit of N is 8.

2. If p(N) = 1 (the minimum value for the product of two digits) and s(N) = 1 (the minimum value for the sum of two digits), then N = 1 + 1 = 2. In this case, the unit digit of N is 2.

So, the possible unit digits for N are 2 and 8, depending on the values of p(N) and s(N).