The number of prime numbers lying between 331 and 345, both included, is:

The correct answer is 2.

To find the number of prime numbers lying between 331 and 345 (both included), we need to check the primality of each integer in the range from 331 to 345.

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
To determine if a number $n$ is prime, we only need to test divisibility by prime numbers up to

$\sqrt{n}$

For $n=345$, since

$\sqrt{345}\approx 18.57$

we only need to check divisibility by prime numbers less than 19: 2, 3, 5, 7, 11, 13, and 17.

Let's check each number in the given range:

1. 331:
- Not divisible by 2 (odd number).
- Not divisible by 3 (sum of digits = 3 + 3 + 1 = 7).
- Not divisible by 5 (does not end in 0 or 5).
- Not divisible by 7 (331 = 7 × 47 + 2).
- Not divisible by 11 (331 = 11 × 30 + 1).
- Not divisible by 13 (331 = 13 × 25 + 6).
- Not divisible by 17 (331 = 17 × 19 + 8).
Thus, 331 is a prime number.

2. 332: Divisible by 2 (even) ⇒ Composite.
3. 333: Divisible by 3 (sum of digits = 9) ⇒ Composite.
4. 334: Divisible by 2 (even) ⇒ Composite.
5. 335: Divisible by 5 (ends in 5) ⇒ Composite.
6. 336: Divisible by 2 (even) ⇒ Composite.

7. 337:
- Not divisible by 2 (odd number).
- Not divisible by 3 (sum of digits = 3 + 3 + 7 = 13).
- Not divisible by 5 (does not end in 0 or 5).
- Not divisible by 7 (337 = 7 × 48 + 1).
- Not divisible by 11 (337 = 11 × 30 + 7).
- Not divisible by 13 (337 = 13 × 25 + 12).
- Not divisible by 17 (337 = 17 × 19 + 14).
Thus, 337 is a prime number.

8. 338: Divisible by 2 (even) ⇒ Composite.
9. 339: Divisible by 3 (sum of digits = 15) ⇒ Composite.
10. 340: Divisible by 2 (even) ⇒ Composite.
11. 341: Divisible by 11 (341 = 11 × 31) ⇒ Composite.
12. 342: Divisible by 2 (even) ⇒ Composite.
13. 343: Divisible by 7 (343 = 7 × 49 = 7³) ⇒ Composite.
14. 344: Divisible by 2 (even) ⇒ Composite.
15. 345: Divisible by 5 (ends in 5) ⇒ Composite.

Therefore, the only prime numbers in the given range are 331 and 337, giving a total of 2 prime numbers.