To find the largest prime number under 1,000, we start by identifying the largest 3-digit number, which is 999, and check for the largest prime below this number.
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. To be classified as a prime, a number must have exactly two distinct positive divisors: 1 and itself.
Let's examine the largest numbers below 1,000, starting with 999, and checking for primality:
- 999 - This number is even and can be divided by 3 (999 = 3 x 333); hence it is not prime.
- 998 - This number is also even; hence it is not prime.
- 997 - Now we check if 997 is a prime number.
To test for primality, we can perform trial division using prime numbers less than the square root of 997. The approximate square root of 997 is around 31.56, so we test divisibility with prime numbers less than or equal to 31: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31.
- 2: 997 is odd, so it is not divisible.
- 3: The sum of the digits \(9 + 9 + 7 = 25\) is not divisible by 3.
- 5: The last digit is 7, which is not 0 or 5.
- 7: \(997 \div 7 \approx 142.43\) (not an integer).
- 11: \(997 \div 11 \approx 90.64\) (not an integer).
- 13: \(997 \div 13 \approx 76.69\) (not an integer).
- 17: \(997 \div 17 \approx 58.65\) (not an integer).
- 19: \(997 \div 19 \approx 52.47\) (not an integer).
- 23: \(997 \div 23 \approx 43.35\) (not an integer).
- 29: \(997 \div 29 \approx 34.79\) (not an integer).
- 31: \(997 \div 31 \approx 32.16\) (not an integer).
Since 997 is not divisible by any prime number up to 31, it does not have any divisors other than 1 and itself. Therefore, 997 is indeed a prime number.
Hence, the largest prime number under 1,000 is
\[ \boxed{997} \]