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basic10-6

# basic10-6 - UIUC Putnam Training Sessions Fall 2010 Putnam...

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UIUC Putnam Training Sessions Fall 2010 Putnam Training Session 6: Number Theory Primes and Composite Numbers: An integer n 2 is called prime if its only positive divisors are 1 and n ; and is called composite otherwise. Equivalently, an integer n 2 is prime if it cannot be written in the form n = ab with integers a, b 2. (Note that only integers 2 are classified as prime or composite; in particular, the number 1 is neither prime nor composite.) Congruences: Let a, b Z , and m N . We say a is congruent to b modulo m , and write “ a b mod m ”, if m divides a - b . The integer m is called the modulus of the congruence. Properties of congruences What makes congruences so useful is that, to a large extent, they can be manipulated like ordinary equa- tions. Congruences to the same modulus can be added, multiplied, and taken to a fixed positive integral power; i.e., for any a, b, c, d Z and m N we have: 1. If a b mod m and c d mod m , then a + c b + d mod m .

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