What is a test for finding a prime number?

Answer 1

Divide it by prime numbers that are less than or equal to the square root of the number. If any of the smaller prime numbers evenly divide into the original number, then the original number is not prime. Otherwise, the original number is prime.

Example: Determine if 253 is prime.

Solution: First determine that the square root of 253 is about 15.9. So we check all the prime numbers less than or equal to 15 (namely: 2, 3, 5, 7, 11, and 13) to see if any of them evenly divide into 253. The number 11 does divide evenly into 253, so 253 is not prime.

Example: Determine if 659 is prime.

Solution: The square root of 659 is about 25.7, so we check all the prime numbers less than or equal to 25 (namely: 2, 3, 5, 7, 11, 13, 17, 19, and 23) to see if any of them evenly divide into 659. None of them do, so 659 is prime.

Example: Determine if 1073 is prime.

Solution: The square root of 1073 is about 32.8, so we check all the prime numbers less than or equal to 32 (namely: 2, 3, 5, 7, 11, 13, 17,19, 23, 29, and 31) to see if any of them evenly divide into 1073. The number 29 does, so 1073 is not prime.

TIP: When dividing large numbers by the smaller prime numbers there are easy ways to see if some are divisible without a calculator.

If the number is even it is divisible by 2.

If the digits in it add up to a number that is divisible by three than it is divisible by 3.

ex. 234 2+3+4 = 9, and 9 is divisible by 3. So 234 is also divisible by 3.

If the number ends in 0 or 5 it is divisible by 5.

To test for divisibility by 11, alternately add and subtract each digit in the number. If the result is divisible by 11, then the original number is divisible by 11; otherwise it is not.

ex. 43,795,293,718 is divisible by 11, since if you add 4 (starting of course at zero), then subtract 3, then add 7, then subtract 9, then add 5, then subtract 2, then add 9, then subtract 3, then add 7, then subtract 1, and finally add 8, the result is 22, which is divisible by 11.