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How to test if a number is prime up to 1000

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Detailed Solution

For numbers up to 1,000, the most efficient manual method is Trial Division with an optimization.

The key optimization is that you only need to check for factors up to the square root of the number (n). If n has a factor larger than its square root, it must also have a factor smaller than it, which you would have already found.

Let's test if n = 97 is prime. The square root of 97 is ~9.8. We only need to check primes up to 9 (which are 2, 3, 5, 7).

Step 1: Is n <= 1? No. (97 is not 1).

Step 2: Is n = 2? No. (It's 97).

Step 3: Is n divisible by 2? 97 / 2 has a remainder. No.

Step 4: Check odd divisors from 3 up to sqrt(n), which is ~9.8. We check 3, 5, 7, 9.

Is 97 divisible by 3? 9+7=16. 16 is not divisible by 3. No.
Is 97 divisible by 5? It doesn't end in 0 or 5. No.
Is 97 divisible by 7? 97 / 7 = 13 with a remainder of 6. No.

Step 5: We have checked all prime divisors up to the square root of 97. Since none of them divided it evenly, 97 is prime.

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