Solution:
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.
Visit
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 find the sum of all n’s if n are integers between 1 and 100, inclusive.
Follow the second and the third steps: From the original condition, we have 1 variable (n). To match the number of variables with the number of equations, we need 1 more equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.
Recall 3 Principles and choose D as the most likely answer.
Let’s look at each condition separately.
Condition (1) tells us that \(2^n\ \) - 1 is a multiple of 5.
=> Multiples of 5 ends either with 0 or 5 and Powers of 2 ends with 2, 4, 8, or 6.
=> \(2^n\ \) - 1 will be a multiple of 5 when the power of 2 ends with a 6. This is only possible when 2 is raised to a power of 4 or a multiple of 4. For example: n = 4 = \(2^n\ \) - 1 = 16 - 1 = 15 (multiple of 5).
In other words, n should be multiple of 4.
From 1 to 100, we have 24 multiples of 4, starting with 4 and ending with 100. Therefore n = 24.
=> The sum of the 24 numbers will be \(\frac{n}{2}\) * [first term + last term]
=> \(\frac{25}{2}\) * [4 + 100]
=> 1,300
Since the answer is unique, the condition is sufficient, according to CMT 2, which states that the number of answers must be one.
Condition (2) tells us that n is a multiple of 4.
=> From 1 to 100, we have 24 multiples of 4, starting with 4 and ending with 100. Therefore n = 24.
The sum of the 24 numbers:
=> \(\frac{n}{2}\) * [first term + last term]
=> \(\frac{25}{2}\) * [4 + 100]
=> 1,300
Since the answer is unique, the condition is sufficient, according to CMT 2, which states that the number of answers must be one.
EACH condition ALONE is sufficient.
Therefore, D is the correct answer.
Also, according to Tip 1, if both the conditions give the same value, the most probable answer is D. It is about 95% likely that D would be the answer when the value of the Condition (1) is equal to the value of the Condition (2).
The answer is D because (1) = (2).
EACH condition ALONE is sufficient.
Therefore, D is the correct answer.
Answer: D
