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Unformatted text preview: a qb r = + with b < Examples: Why is the division algorithm useful? Show that the square of any integer has the form 4k or 4k+1. Note that if we want to prove something about the divisibility of integers, we cant test all of them. But, if an integer is divided by n how many possible remainders are there? Math 4383 Section 2.2 Page 3 of 3 In the case of n = 7 EVERY INTEGER is of EXACTLY ONE of the following forms: If I want to prove that the cube of every integer is of the form 7k or 7k+1 Note that if we define a relation on the set of integers defined by x is related to y if they have the same remainder when divided by n this is an equivalence relation. The distinct equivalence classes are indexed by the remainders 0 n-1. EVERY integer falls into EXACTLY ONE of these equivalence classes (the relation PARTITIONS the set of integers)....
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