ExamSoln - MATH 310 MIDTERM EXAM SOLUTIONS 1(a State the...

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MATH 310, MIDTERM EXAM SOLUTIONS 1. (a) State the definition of Z n . Answer: The set Z n is obtained from Z by changing the definition of the “equals” sign as follows: a = b in Z n if and only if a = b + nq for some q Z . Another way to express this idea is to draw n points in a circle, name one of them “0”, and then name each point by the number of (clockwise, or counter- clockwise) steps from “0.” (b) Explain how to “add” and “multiply” elements of Z n . Answer: Recall that every element of Z n has infinitely many names. Given two elements of Z n , we can “add” them by adding their names. This process is well-defined because the result does not depend on which names we choose for each element: If a = b in Z n and c = d in Z n , then a + c = b + d in Z n . Similarly, we can “multiply” two elements of Z n by multiplying their names, and this is well-defined because: If a = b in Z n and c = d in Z n , then ac = bd in Z n .
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