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Closure of a set of even numbers | TOMATO objective 27

Try this beautiful problem from TOMATO Objective no. 27 based on Closure of a set of even numbers.

Problem: Closure of a set of even numbers

S is the set whose elements are zero and all even integers, positive and negative. Consider the 5 operations- [1] addition;  [2] subtraction;   [3] multiplication; [4] division; and [5] finding the Arithmetic Mean. Which of these operations applied to any pair of elements of S, yield only elements of S.


Adding, subtracting and multiplying even numbers  yield even numbers.

Division may yield non-integer (\frac {2}{4} = \frac{1}{2} ) or odd number (\frac {12}{4} = 3 ) both of which are not members of S. So division is ruled out.

Finally arithmetic of two even numbers may be a odd number. For example, \frac { 4 + 6}{2} = 5 .

Thus operations [1], [2], [3] work.

Answer: (D) [1], [2], [3]

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