A number theory class I once took provided us with the following "proof", claiming, in fact, that it was the most useful thing we would learn, and would make a great party trick.
Theorem: Anything less than 91 which looks prime is prime.
Proof: To do this, we will calculate the smallest number that looks prime but isn't. The numbers 2, 3, 4, 5, and 6 all have easy and quick divisibility rules, so our number cannot be divisible by any of them. 7, however, is hard to check for - so clearly the first number that looks prime but isn't will be divisible by 7. Sadly, everyone realizes 49 isn't prime, because it's a square, so we need to find the next difficult prime factor. 8, 9, 10, 11, and 12 are all also subject to quick tests and divisibility rules, so discount those. 13, however, is ugly - thus, the first number that looks prime but isn't is 7 times 13, which equals 91. Q.E.D. (My professor insisted this stood for "Quite Easily Done".)
Theorem: Anything less than 91 which looks prime is prime.
Proof: To do this, we will calculate the smallest number that looks prime but isn't. The numbers 2, 3, 4, 5, and 6 all have easy and quick divisibility rules, so our number cannot be divisible by any of them. 7, however, is hard to check for - so clearly the first number that looks prime but isn't will be divisible by 7. Sadly, everyone realizes 49 isn't prime, because it's a square, so we need to find the next difficult prime factor. 8, 9, 10, 11, and 12 are all also subject to quick tests and divisibility rules, so discount those. 13, however, is ugly - thus, the first number that looks prime but isn't is 7 times 13, which equals 91. Q.E.D. (My professor insisted this stood for "Quite Easily Done".)