MathRevolution wrote:
Que: For a positive integer x, what is the value of the hundreds digit of \(30^x\)?
(1) x ≥ 3.
(2) \(\frac{x}{ 3}\) is an integer.
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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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 the hundreds digit of \(30^x\) - where ‘x’ is a positive integer
Follow the second and the third step: From the original condition, we have 1 variable (x). To match the number of variables with the number of equations, we need 1 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.
Condition (1) tells us that x ≥ 3
Since x ≥ 3, the number of trailing zeroes in \(30^x\) must be at least ‘three.’
For example,
=> For x = 3, \(30^x\) = \(30^3\) = 27,000
=> For x = 4, \(30^x\) = \(30^4\) = 810,000
Thus, the hundreds digit of \(30^x\) is ‘0’.
The answer is a unique value; condition (1) alone is sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.
Condition (2) tells us that \(\frac{x }{3}\) is an integer.
Thus, the possible values of x are 3, 6, 9 . . .
=> For x = 3, \(30^x\) = \(30^3\) = 27,000
=> For x = 6, \(30^x\) = \(30^6\) = 729,000,000
Thus, the hundreds digit of \(30^x\) is ‘0’.
The answer is a unique value; condition (2) alone is sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.
Also, according to Tip 1, it is about 95% likely that D is the answer when condition (1) = condition (2).Each condition alone is sufficient.
Therefore, D is the correct answer.
Answer: D