A number is prime if its only factors are 1 and itself. There is no easy way to check primes larger than about 100, but there are programs online that will check for you. A large prime can only end in 1, 3, 7, or 9.*
The quick checks that indicate a number is NOT prime:
For primes less than 100, you can also check to see if the number is divisible by 7, 11, or 13.
* (See the two related links for large Prime number checking sites)
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A prime number can only be divided by itself and 1
Multiply every combination of numbers possible. If nothing equals the number you want to know is prime, it's prime. One quick check you can make: if the number is 2 digits or more and it ends in 0, 2, 4, 5, 6 or 8, then it most definitely is NOT prime. For all other numbers, you have to do some calculating to be sure.
Find a prime number, add 2 to the number. Check if the new number is prime. IE : 3 is prime. 3+2 =5. 5 is prime. (3,5) are twin primes.
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There is no quick way to work out if a number is prime or not, the only way is to see if it can be divided by any numbers that aren't itself or one.