Let both p and k be prime numbers such that (p² + k) is also a prime number less than 30. What is the number of possible values of k?

The correct option is 5 (which corresponds to Option 5 in the list of choices).

Step 1: Understand the given condition

We are given that $p$ and $k$ are both prime numbers.

We are told that $p^2+k$ is also a prime number, and this prime number must be strictly less than 30.

Step 2: Determine possible values for $p$

Since $p$ and $k$ are prime numbers, the smallest prime number is 2.

If $p\ge 5$, then $p^2\ge 25$. Since $k\ge 2$, $p^2+k\ge 27$. Let's check candidate values of $p$:

1. If $p=2$:
$p^2=4$
We need $4+k$ to be a prime number less than 30, where $k$ is a prime number.
Let's check possible prime values for $k$:
- If $k=3$: $4+3=7$ (Prime < 30) ⇒ Valid! ($k=3$)
- If $k=7$: $4+7=11$ (Prime < 30) ⇒ Valid! ($k=7$)
- If $k=9$ (Not prime)
- If $k=13$: $4+13=17$ (Prime < 30) ⇒ Valid! ($k=13$)
- If $k=19$: $4+19=23$ (Prime < 30) ⇒ Valid! ($k=19$)
- If $k=23$: $4+23=27$ (Not prime, $27=3\times 9$)
- Other primes for $k$ like 2, 5, 11, 17: $4+2=6$ (even), $4+5=9$ (not prime), $4+11=15$ (not prime), $4+17=21$ (not prime).

2. If $p=3$:
$p^2=9$
We need $9+k$ to be a prime number less than 30, where $k$ is prime.
- If $k=2$: $9+2=11$ (Prime < 30) ⇒ Valid! ($k=2$)
- If $k=4$ (Not prime)
- If $k=8$ (Not prime)
- If $k=10$ (Not prime)
- For any odd prime $k$, $9+k$ will be even and greater than 2, hence composite. So only $k=2$ works when $p=3$.

3. If $p=5$:
$p^2=25$
We need $25+k<30$, so $k<5$.
- If $k=2$: $25+2=27$ (Not prime)
- If $k=3$: $25+3=28$ (Not prime)

4. If $p\ge 7$:
$p^2\ge 49$, which already exceeds 30.

Step 3: Collect all unique valid values of $k$

The unique prime values of $k$ that satisfy the condition are:

$k\in \{2,3,7,13,19\}$

Counting the total number of valid values of $k$, we get exactly 5 values.

Thus, the number of possible values of $k$ is 5.