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PMATH340 LN8

# PMATH340 LN8 - Chapter 4 Congruences For this section it is...

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Chapter 4 Congruences For this section, it is helpful if you keep “clock arithmetic” in mind. That is 14:00 hours is 2 o’clock. This will be considered doing arithmetic modulo 12. We will just consider an n hour clock. Definition 5. Let n N . Two integers a and b are said to be congruent modulo n , written a b (mod n ) , provided n | ( a b ). The canonical example of a congruence comes from the division algorithm. Example 5. Let a Z and n N . Given the division algorithm we know that there exists unique q, r Z with 0 r < n , such that a = qn + r. The definition of congruence gives a r (mod n ) . Definition 6. If a, b Z and a b (mod n ) we say that a is a residue of b modulo n . Remark 4. The above examples tells us that modulo n any integer a is con- gruent to one of 0 , 1 , 2 , . . . , n 1 . These integers are called the least nonnegative residues modulo n . Definition 7. The set of integers { a 0 , a 1 , . . . , a n 1 } is called a complete residue system modulo n provided (i) a i a j (mod n ), whenever i = j (ii) for each integer k there corresponds an a i such that k a i (mod n ).

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