Identifying Prime Numbers Between 389 and 403

To find the number of prime numbers between 389 and 403, we need to check the primality of each integer in this range (inclusive).

A prime number is a natural number greater than 1 that has only two distinct positive divisors: 1 and itself.

We need to test numbers from 389 to 403.

Primality Testing Procedure

To check if a number '$n$' is prime, we test its divisibility by prime numbers up to the square root of '$n$'.

For the upper bound, 403, the square root is approximately $\sqrt{403} \approx 20.07$.

The prime numbers less than 20.07 are 2, 3, 5, 7, 11, 13, 17, and 19.

Checking Numbers in the Range

• 389: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19. Prime.
• 390: Divisible by 2 and 5. Not prime.
• 391: $391 = 17 \times 23$. Not prime.
• 392: Divisible by 2. Not prime.
• 393: Divisible by 3 ($3+9+3=15$). Not prime.
• 394: Divisible by 2. Not prime.
• 395: Divisible by 5. Not prime.
• 396: Divisible by 2 and 3. Not prime.
• 397: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19. Prime.
• 398: Divisible by 2. Not prime.
• 399: Divisible by 3 ($3+9+9=21$) and 7 ($399 = 7 \times 57$). Not prime.
• 400: Divisible by 2 and 5. Not prime.
• 401: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19. Prime.
• 402: Divisible by 2 and 3. Not prime.
• 403: $403 = 13 \times 31$. Not prime.

Final Count

The prime numbers identified in the range [389, 403] are 389, 397, and 401.

Therefore, there are 3 prime numbers between 389 and 403, inclusive.