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Re: If M is a finite set of negative integers, is the total number of inte
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29 Dec 2015, 23:14
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution
If M is a finite set of negative integers, is the total number of integers in M an odd number?
(1) The product of all the integers in M is odd
(2) The product of all the integers in M is negative
There are numerous variables (because we have to know the total number of integers in M) in the original condition. In order to match the number of variables and the number of equations, we need numerous equations as well. Since the condition 1) and 2) each has 1 equation, there is high chance that E is going to be the answer. Using both the condition 1) and 2), we can see that the total number of integers in M has to be an odd number in order for the product of all the integers in M to be odd and negative. Therefore, the answer is ‘yes’ and the correct answer is C. This is an integer question, which is one of the key questions. This means we have to apply the Common Mistake Type 4(A).
In case of the condition 1), it states that the product of all the integers in M is odd and there is no way we can know if the total number of integers in M is an odd number. In other words if M={-1, -3}, then the answer is ‘no’. If M={-1, -3, -5}, then the answer is ‘yes’. Therefore, the condition is not sufficient.
In case of the condition 2), in order for the product of all the integer in M to be negative, the total number of integers in M has to an odd number. Therefore, the answer is ‘yes’ and the condition is sufficient. The answer is, then, B.
For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.