5. Alessandro, Daniele, and Manuela are discussing a two-digit natural number $n$. Each of them makes two statements, but since they are all a bit poor at math, each of them makes one true statement and one false statement.

Alessandro says: " $n$ is even. Moreover, it is a multiple of 3.";

Daniele responds: "Yes, $n$ is a multiple of 3. Furthermore, the units digit of $n$ is 5.";

Manuela says, finally: " $n$ is a multiple of 5. The sum of its digits is 12.".

How many values can $n$ take?
(A) No such $n$ exists.
(B) 1
(C) 2
(D) 3
(E) 4
5. The answer is (D). Suppose first that $n$ is even: then it is not a multiple of 3, because one of the two statements made by Alessandro must be false, and therefore (using what Daniele says) the units digit of $n$ must be 5, but this is impossible for an even number. The number $n$ (if it exists) is therefore a multiple of 3: the first statement made by Alessandro is then false, so $n$ is odd, and according to what Daniele says, the units digit cannot be 5. Therefore, $n$ cannot be a multiple of 5, because its units digit (which cannot be 0, since $n$ is odd) would be 5: it follows that the true statement made by Manuela is that the sum of the digits is 12. The only two-digit numbers whose digits add up to 12 are $39, 48, 57, 66, 75, 84$ and 93, and among these we must take only those that are odd and multiples of 3, which are 39, 57 and 93.