$P, Q, R, S$, and $T$ are five different integers between 2 and 19 inclusive.

- $P$ is a two-digit prime number whose digits add up to a prime number.
- $Q$ is a multiple of 5 .
- $R$ is an odd number, but not a prime number.
- $S$ is the square of a prime number.
- $T$ is a prime number that is also the mean (average) of $P$ and $Q$.

Which number is the largest?
(A) $P$
(B) $Q$
(C) $R$
(D) $S$
(E) $T$
Let us first consider the possibilities for each integer separately:

- The two-digit prime numbers are $11,13,17,19$. The only one whose digits add up to a prime number is 11 . Therefore, $P=11$.
- Since $Q$ is a multiple of 5 between 2 and 19, then the possible values of $Q$ are $5,10,15$.
- The odd numbers between 2 and 19 that are not prime are 9 and 15 , so the possible values of $R$ are 9 and 15 .
- The squares between 2 and 19 are 4, 9 and 16 . Only 4 and 9 are squares of prime numbers, so the possible values of $S$ are 4 and 9 .
- Since $P=11$, the possible value of $Q$ are 5,10 and 15, and $T$ is the average of $P$ and $Q$, then $T$ could be $8,10.5$ or 13 . Since $T$ is also a prime number, then $T$ must be 13, so $Q=15$.

We now know that $P=11, Q=15$ and $T=13$.

Since the five numbers are all different, then $R$ cannot be 15 , so $R=9$.

Since $R=9, S$ cannot be 9 , so $S=4$.

Therefore, the largest of the five integers is $Q=15$.

ANsWER: (B)