College Algebra Exam Review 28

# College Algebra Exam Review 28 - b modulo n ” and we...

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38 1. ALGEBRAIC THEMES line, with the integer points marked, and wrapping it around the circum- ference of the clock, with 0 on the number line coinciding with 0 on the clock face. Then the numbers :::; ± 3n; ± 2n; ± n;0;n;2n;3n;::: on the number line all overlay 0 on the clock face. The numbers :::; ± 3n C 1; ± 2n C 1; ± n C 1;1;n C 1;2n C 1;3n C 1;::: on the number line all overlay 1 on the clock face. In general, for 0 ² k ² n ± 1 , the numbers :::; ± 3n C k; ± 2n C k; ± n C k;k;n C k;2n C k;3n C k;::: on the number line all overlay k on the clock face. When do two integers on the number line land on the same spot on the clock face? This happens precisely when the distance between the two numbers on the number line is some multiple of n , so that the interval between the two numbers on the number line wraps some integral number of times around the clock face. Since the distance between two numbers a and b on the number line is j a ± b j , this suggests the following deﬁnition: Deﬁnition 1.7.1. Given integers a and b , and a natural number n , we say that “ a is congruent to
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Unformatted text preview: b modulo n ” and we write a ³ b . mod n/ if a ± b is divisible by n . The relation a ³ b . mod n/ has the following properties: Lemma 1.7.2. (a) For all a 2 Z , a ³ a . mod n/ . (b) For all a;b 2 Z , a ³ b . mod n/ if, and only if, b ³ a . mod n/ . (c) For all a;b;c 2 Z , if a ³ b . mod n/ and b ³ c . mod n/ , then a ³ c . mod n/ . Proof. For (a), a ± a D is divisible by n . For (b), a ± b is divisible by n if, and only if, b ± a is divisible by n . Finally, if a ± b and b ± c are both divisible by n , then also a ± c D .a ± b/ C .b ± c/ is divisible by n . n For each integer a , write ŒaŁ D f b 2 Z W a ³ b . mod n/ g D f a C kn W k 2 Z g : Note that this is just the set of all integer points that land at the same place as a when the number line is wrapped around the clock face. The set ŒaŁ is called the residue class or congruence class of a modulo n ....
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