Solution: To save time and improve accuracy on DS questions in GMAT, learn, and apply Variable Approach.
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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https://www.mathrevolution.com/gmat/lesson for details.
Now we will solve this DS question using the Variable Approach.
Let’s apply the 3 steps suggested previously.
Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
We have to verify whether 'p' and '10' have a common factor other than 1 – where ‘m’, ‘n’, and ‘p’ are positive integers and 4m + 5n = p.
For 'p' and '10' to have a common factor other than 1, since 5n is a multiple of 5, 4m should be divisible by 5 or m should be divisible by 5.
Thus, look at condition (1), it tells us that m is a multiple of 5.
Assume m=5t where 't' is an integer, we get
=> 4m + 5n = 4(5t) + 5n = 5(4t + n) = p, so p has a factor of 5.
Thus, p and 10 have a common factor other than 1, which is equal to 5, so we get YES as an answer.
The answer is unique, yes, so condition (1) alone is sufficient according to Common Mistake Type 1 which states that the answer should be a unique YES or a NO.
Condition (2) tells us that n is a multiple of 5, from which we cannot determine whether p and 10 have a common factor other than 1.
For example, if n=5 and m=5, then p = 4m + 5n = 4(5) + 5(5) = 45.
And 'p' and '10' have a common factor other than 1, which is equal to 45 and we get YES as an answer.
However, if n=5 and m=1, then p = 4m + 5n = 4(1) + 5(5) = 29. And p and 10 have not a common factor other than 1, which is equal to 1 and we get NO as an answer.
The answer is not unique, YES or NO, so condition (2) alone is not sufficient according to Common Mistake Type 1 which states that the answer should be a unique YES or a NO.
Condition (1) alone is not sufficient.
Therefore, A is the correct answer.
Answer: A
