How do you test if a number is composite?

To test if a number is composite, you need to find out if it has any factors other than 1 and itself. If you can find just one, the number is composite.

The most reliable method is trial division:

1. Pick a number you want to test (let's call it N).
2. Start by checking if N is divisible by 2. If it is (and N is not 2 itself), it's composite.
3. If not, try dividing N by the next prime number, 3. If it divides evenly, it's composite.
4. Continue testing with the next prime numbers (5, 7, 11, 13, and so on).

You can also use divisibility rules as shortcuts. For example:

• If a number ends in 0 or 5, it's divisible by 5 (and therefore composite, unless it's 5 itself).
• If a number is even (ends in 0, 2, 4, 6, 8), it's divisible by 2 (and composite, unless it's 2 itself).
• If the sum of its digits is divisible by 3, the number is divisible by 3 (and composite, unless it's 3 itself).

Important Tip: You only need to check for prime factors up to the square root of the number N. For example, to test if 97 is composite, you only need to check for prime factors up to $\sqrt{97}$ (which is about 9.8). So, you only need to test 2, 3, 5, and 7. Since none of them divide 97 evenly, 97 is prime, not composite.