Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For

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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 could N be? Is there more than one answer?

For example, for 23 P(23) = 6 and S(23) = 5, but 23 could not be the N that we want since 23 <> 5 + 6

Let t = tens digit and o = ones digit
P(n) = to
S(n) = t + o
P(n) + S(n) = to + t + o
N = 10t + o

Set them equal to each other N = P(N) + S(N)
10t + o = to + t + o

o's cancel, so we have
10t = to + t

Subtract t from each side, we have
9t = to

Divide each side by t
o = 9

So any two-digit number with 9 as the ones digit will work:
{19,29,39,49,59,69,79,89,99}