13) Giulio wants to surprise Damiano with his divinatory skills. For this, he provides him with a list of some two-digit numbers and asks him to choose one. Giulio asks Damiano the sum of the digits of the number, and is so sure he can guess it. At most, how many numbers was the initial list composed of?
(A) 9
(B) 10
(C) 11
(D) 17
(E) 18.
13) The answer is $(\mathbf{E})$. To be certain of deriving the number Damiano has chosen from the sum of the digits, Giulio must have written at most one number for each possible sum (so that the sum of the digits of each of them is different from that of all the others). The sum of the digits of a two-digit number can only take on the integer values from 1 to 18; indeed, such a sum is minimal if one considers the number 10 (in which case the sum is 1), and it is maximal if one considers the number 99 (in which case the sum is 18). It is easy to verify that all values between 1 and 18 are obtainable, so it follows that there are 18 possible distinct sums that can be derived from two-digit numbers, and therefore 18 is the maximum number of numbers that Giulio could have written in the list.