A prime number, by definition, is a number that has only itself and 1 as factors. So to tell if a number is prime or not, you need to check and see if it has any other factors besides 1 and itself. The number 465 is divisible by 5, so it isn't a prime number. The number 822 is divisible by 2, so it isn't a prime number. The number 600 is divisible by 10, so it is not a prime number. The number 743 has to be a prime number, because there are no other factors besides 1 and 743 that go into it. Also, you know it is a prime number because it is the only number left that is NOT composite*.
*composite is the opposite of prime