Data Sufficiency

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the value of m, if m is a three-digit positive integer.

Follow the second and the third step: From the original condition, we have 3 variables (3-digit integer). To match the number of variables with the number of equations, we need 3 equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer.

Let’s look at both conditions together.

Conditions (1) and (2) tell us that the digits of m are 5, 6, and 7 without repetition, and m is a product of two consecutive positive integers, from which we get m = 756.

The possible values of m are 567, 576, 657, 675, 756, and 765 from condition (1).
We must find the prime factors of the possible values, which are 567 = 3^4·7, 576 = 2^6·3^2, 657 = 3^2·73, 675 = 3^3·5^2, 756 = 2^2·3^3·7, and 765 = 3^2·5·17.

Then we see that 756 = 22·33·7 = 27·28, which is a unique number as a product of two consecutive integers. The answer is unique, so both conditions (1) and (2) combined are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one. So, C seems to be the answer.

However, since this question is a hidden integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately,

Condition (1) tells us that the digits of m are 5, 6, and 7 without repetition, from which we get that the possible values of m are 567, 576, 657, 675, 756, and 765.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition (2) tells us that m is a product of two consecutive positive integers, from which we get that m = 132 since m = 11·12 and m = 156 since m = 12·13.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, both conditions (1) and (2) combined are sufficient.
Both conditions (1) and (2) together are sufficient.

Therefore, C is the correct answer.
Answer: C

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.