Let P(n) and S(n) denote the product ad 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 units digit of N?

Question

Let P(n) and S(n) denote the product ad 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 units digit of N?

Answer 1

Let's assume the two-digit number N can be written as a = 10x + y, where x is the tens digit and y is the units digit.

Given that N = P(N) + S(N), we can rewrite this as:
10x + y = P(10x + y) + S(10x + y)

Using our definitions of P(n) and S(n), we have:
10x + y = (x)(y) + (x + y)

Expanding and simplifying, we get:
10x + y = xy + x + y

Rearranging terms, we have:
9x = xy + x

Dividing both sides by x, we have:
9 = y + 1

Simplifying further, we find:
y = 8

Therefore, the units digit of N is 8.