Max@Math Revolution wrote:[GMAT math practice question]

If m and n are positive integers, what is the value of m + n?

1) m^2< 5
2) m - n = 1

Statement 1:
Since m must be a positive integer, only two cases are possible:
Case 1: m=1, with the result that m² = 1
Case 2: m=2, with the result that m² = 4
Since the value of n is unknown, INSUFFICIENT.

Statement 2:
m = n+1
Case 1: n=1, with the result that m=2 and m+n = 3
Case 2: n=2, with the result that m=3 and m+n = 5
Since m+n can be different values, INSUFFICIENT.

Statements combined:
Only the case in blue satisfies both statements.
Thus, m+n = 3.
SUFFICIENT.

The correct answer is C.

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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.

Since we have 2 variables (m and n) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Condition 1) tells us that m = 1 or m = 2.
If m = 1, then n = 1 - 1 = 0, which is not a positive integer.
If m = 2, then n = 2 - 1 = 1.
Thus, the unique values are m = 2 and n = 1, and m + n = 3.
Both conditions together are sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since it provides no information about the variable n, condition 1) is not sufficient.

Condition 2)
If m = 2 and n = 1, then m + n = 3.
If m = 3 and n = 2, then m + n = 5.
Since we don't have a unique solution, condition 2) is not sufficient.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.