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

Discussion in 'Calculator Requests' started by math_celebrity, Jan 8, 2017.

  1. math_celebrity

    math_celebrity Administrator Staff Member

    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}
     

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