Prime numbers: How would you do this efficiently?

Which of the following integers is prime: 187, 287, 387, 487, or 587? I can calculate it by hand, but that would take a long time. Is there an easier way? I noticed the numbers only differ 100 from each other, but can I use that fact? Exactly one of these must be prime.

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Is it given that exactly one of these numbers must be prime? – Myself Jan 31 '12 at 11:32
Sorry, forgot that fact, that's true. – Kevin Jan 31 '12 at 11:44
The Divisibility rules will be helpful in this case. – userxxxxx Jan 31 '12 at 12:56
Also, the question is not correct, since two of these numbers are prime (factmonster.com/math/numbers/prime.html). – userxxxxx Jan 31 '12 at 13:05
– Frank Science Jan 31 '12 at 14:27

2 Answers

187 is obviously a multiple of 11. 287 is obviously a multiple of 7. 387 is obviously a multiple of 3. So you just have to work on 487 and 587.

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Also, if a number is not prime, one factor is always less than or equal to the square root of that number

For example, the square root of 487 is a little more than 22. So you need only check the primes below 22 (and you see immediately that 2,3 or 5 doesn't work, so you need only check 5 primes).

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Moreover, you don't have to try 7, since it divides 287 but not the difference between 487 and 287; for similar reasons, you don't have to check 11 or 17. Adding 13 to 487 gives 500, which is not a multiple of 13, so that's out. 57 is a multiple of 19, but 430 isn't, so 487 isn't. – Gerry Myerson Jan 31 '12 at 12:10