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The symbol # represents one of the four arithmetic operations: addition, subtraction, multiplication, and division. Is (5 # 6) # 2 = 5 # ( 6 # 2)?
(1) 5 # 6 = 6 # 5
(2) 2 # 0 = 2
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Since we have one operation(#), we can assume we have 1 variable (#) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.
Condition 1)
The possible operations are + and x for condition 1), because 5+6=6+5, 5x6=6x5, but 5-6≠6-5, 5÷6≠6÷5
Both operations have associative properties and we have (5#6)#2=5#(6#2)
The answer is 'yes'.
Since condition 1) yields a unique solution, it is sufficient
Condition 2)
The possible operations are + and - for condition 1), because 2+0=2, 2-0=2, but 2x0≠2, 2÷0 is not defined.
If # is an addition, then we have (5+6)+2=5+(6+2) and the answer is 'yes'.
If # is a subtraction, then we have (5-6)-2≠5-(6-2) and the answer is 'no'.
Since condition 2) does not yield a unique solution, it is not sufficient
Therefore, A is the answer.
If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.