MathRevolution wrote:Q. (Number) What is the value of a positive integer n?
1) 756n + 576 is a perfect square number.
2) n is a unit digit number.
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.
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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 find the value of 'n'.
Follow the second and the third step: 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 756n + 576 is a perfect square number , from which we get as below,
756n + 576 = (2*2*3*3*3*7)n + 2*2*2*2*2*2*3*3
= (36*21)n + 36*16 = 36(21n + 16)
Since 756n + 576 is a perfect square, (21n + 16) has to be a perfect square as 36 is already the perfect square of 6. We can find the value of n as shown below:
=> 21n + 16 = 36 => 21n = 20 => n = not an integer.
=> 21n + 16 = 49 => 21n = 33 => n = not an integer.
=> 21n + 16 = 64 => 21n = 48 => n = not an integer.
=> 21n + 16 = 81 => 21n = 65 => n = not an integer.
21n + 16 = 100 => 21n = 84 => n = 4.
21n + 16 = 121 => 21n = 105 => n = 5.
So, the values 4 or 5 are possible for n.
The answer is not unique, so the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one.
Condition (2) tells us that n is a unit digit number, from which it follows that n can take the values from 0 to 9.
The answer is not unique, so the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one.
Let’s look at both conditions together. They tell us that the values of n can be 4 or 5.
The answer is not unique, so both conditions (1) and (2) combined are not sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one.
Both conditions (1) and (2) together are not sufficient.
Therefore, E is the correct answer.
Answer: E 
