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That's a good question. I wouldn't worry too much about it. It's not too tough to check for 3, 7, 11, 17 as you said, and most of the time it's true that if a number looks prime and it's not divisible by 3 then it probably is.
But the right thing to know are the numbers that look prime but aren't. For example, 13x13 is 169. Now think about a number that would be divisible by 13. If it was 13 times any even number, it would be even. If it was 13 times 3 or 9, it would be easy to spot that. So the only things you have to worry about are things like 13x7, or 13x11. Really, we're talking about being prepared to deal with common multiples of prime numbers.
Also, you should know that 441 is 21 squared. You should recognize multiples of 11 from 100-199 (the units digit is ALWAYS one less than the tens digit: 121, 132, 143, 154, etc...).
These numbers come into play much more than really big primes do, and I'd focus on them more than anything else. If you're studying and doing tons of problems, you probably will become familiar with them anyway.
absolutely. but notice that it's divisible by 3 and 9. So no matter what, it wouldn't be prime. And if you weren't sure what to do with it, it breaks down to prime factors 3x3x7x7, which is 3x7x3x7, which is 21x21. So you can always figure it out just that way on the test.
For a quick way to see whether a number is prime, check whether the number is divisible by prime numbers up to the square root of the number (by using divisibility rules).
As an example, say the number you're trying to crack is 461. You need to check all primes up to 19.
Try 2 (first prime): no, not even.
3: no, sum of the numbers not divisible by 3.
5: no, doesn't end with 0 or 5.
7: alternating + or - signs don't end up in number divisible by 7.
11, 13, 17, and 19: easy and don't work either.
So the number is prime. It is a generalization that can be applied to the problem in less than 2 minutes. Get away as much as you can from trial and error - it wastes time and consumes you.
You should also know why we aren't trying numbers like 4, 6, 8...and how we get to the 19 prime limit figure (in this example).