List all prime numbers between 30 and 50 inclusive.

Concept: A prime number is a natural number greater than 1 whose only positive divisors are 1 and itself. To find every prime within a range, test each number for divisibility by every prime not exceeding its square root — if none divide it, the number is prime.

Application: List the numbers from 30 to 50 and eliminate composite numbers systematically.

1. Remove every even number in the range (30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50) — each is divisible by 2, so none can be prime.

2. Remove multiples of 3 among the remaining odd numbers: 33 (3 × 11), 39 (3 × 13), and 45 (3 × 15).

3. Remove multiples of 5: 35 (5 × 7).

4. The candidates left to check are 31, 37, 41, 43, 47, and 49. Since √50 is about 7.07, only divisibility by 2, 3, 5, and 7 needs checking for numbers up to 50.

5. 49 = 7 × 7, so 49 is composite and is removed.

6. None of 31, 37, 41, 43, or 47 is divisible by 2, 3, 5, or 7, so each of them is prime.

Cross-check: Counting from 30 to 50 gives 21 numbers; removing the 11 even numbers, the 3 remaining multiples of 3, the 1 remaining multiple of 5, and 49, leaves exactly five numbers — matching the five primes found above.

Result: the complete list of primes between 30 and 50 inclusive is 31, 37, 41, 43, 47.