M denotes the maximum of x and m the minimum of x.

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[GMAT math practice question]

M denotes the maximum of x and m the minimum of x. What is the integer part of \(\sqrt{M-m}\) ?

1) The integer part of \(\sqrt{3x-2}\) is 9.
2) x is a positive integer.

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Since we have 1 variable (x) and 0 equations, D is most likely the answer. So, we should consider each condition on its own first.

Condition 1)

9 ≤ \(\sqrt{3x-2}\) < 10
=> 81 ≤ 3x - 2 < 100 (squaring)
=> 83 ≤ 3x < 102 (adding 2)
=> 83/3 ≤ x < 102/3 (dividing by 3)

Since x doesn’t have a maximum value, condition 1) is not sufficient.

Condition 2)

Since x doesn’t have a maximum value, condition 2) is not sufficient.

Conditions 1) & 2)
When we consider both conditions together, the possible values of x are 28, 29, …, 33.
Then we have M = 33 and m = 29.
Thus, we have \(\sqrt{M-m}=\sqrt{4}\) = 2.

Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the 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.