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A brief introduction to groups

Source: 1959 AMC 12 Problem 14

August 13, 2013
AMCSetsAMC 12

Problem Statement

Given the set SS whose elements are zero and the even integers, positive and negative. Of the five operations applied to any pair of elements: (1) addition (2) subtraction (3) multiplication (4) division (5) finding the arithmetic mean (average), those elements that only yield elements of SS are: <spanclass=latexbold>(A)</span> all<spanclass=latexbold>(B)</span> 1,2,3,4<spanclass=latexbold>(C)</span> 1,2,3,5<spanclass=latexbold>(D)</span> 1,2,3<spanclass=latexbold>(E)</span> 1,3,5 <span class='latex-bold'>(A)</span>\ \text{all} \qquad<span class='latex-bold'>(B)</span>\ 1,2,3,4\qquad<span class='latex-bold'>(C)</span>\ 1,2,3,5\qquad<span class='latex-bold'>(D)</span>\ 1,2,3\qquad<span class='latex-bold'>(E)</span>\ 1,3,5