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This implies b E aZ. Since b was choosen arbitrarily, this imples I C aZ. The reverse inclusion aZ C I follows from the assumption that I is closed under addition and subtraction. This proves the claim and the lemma. 2 The Euclidean algorithm The following result has been known for thousands of years. The method is named for the Greek mathematician Euclid who decribed it in book 7 of this "Elements" , written about 2300 years ago. For more details on his life, see the award-winning web site [MacOR] .

Since gcd(nb n2 ) = 1 , the result follows. D 1 . 7. CONGRUENCES 65 General Chinese remainder theorem Theorem 1 . 35. (Chinese remainder theorem, general version) Let n 1 > 1 , n2 > 1 , . . , nk > 1 be pairwise relatively prime integers. Let n = n 1 n2 . . nk . Then x = a 1 ( mod n1 ) , x = a2 ( mod n2 ) , . . , x = ak ( mod nk ) ( 1 . 7) has a simultaneous solution x E Z . Furthermore, if x, x ' are two solutions to (1. 7} then x' = x (mod n) . This follows from the k = 2 case proven above using mathematical induc­ tion.

Primality testing In this section, we examine some simple general methods for determining if a number is prime. There are much more efficient (and complicated) primality tests what those which we discuss here 4 - see for example [BS] . The first prime to be discovered having over 1 million digits was the Mersenne prime, 269 72593 - 1. The primality of this integer was determined by Nayan Hajratwala in 1993, who received \$ 50000 from the Electronic Frontier Foundation as a reward for it's discovery.

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