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20 Sep 2022, 00:18
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D
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Difficulty:
55%
(hard)
Question Stats:
63% (02:02) correct
37%
(02:08)
wrong
based on 104
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If x is an integer between 2 and 100 and if √x is also an integer, what is the value of x?
(1) \(\sqrt[3]{x}\) is an integer.
(2) \(2(\sqrt[3]{x}) = \sqrt{x}\)
(1) \(\sqrt[3]{x}\) is an integer.
(2) \(2(\sqrt[3]{x}) = \sqrt{x}\)
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20 Sep 2022, 05:27
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carcass
Solution:
Pre-Analysis:
- x is an integer between 2 and 100
- Next, we are given that \(\sqrt{x}\) is also an integer
- This means x is a perfect square and the possible values of x are 4, 9, 16, 25, 36, 49, 64, and 81
Statement 1: \(\sqrt[3]{x}\) is an integer
- According to this statement, x is a perfect cube
- Among the possible values, that we inferred in the pre-analysis step, of x, only 64 is a perfect cube because \(\sqrt[3]{64}=4\)
- So, the value of x is 64
- Thus, statement 1 alone is sufficient and we can eliminate options B, C and E
Statement 2: \(2(\sqrt[3]{x}) = \sqrt{x}\)
- We already know from the question stem that \(\sqrt{x}\) is an integer
- Which means, according to this statement, \(2(\sqrt[3]{x})\) will also be an integer
- Because \(2(\sqrt[3]{x}) = \sqrt{x}\)
- Now for \(2(\sqrt[3]{x})\) to be an integer, \(\sqrt[3]{x}\) has to be an integer
- We already know from statement 1 that if \(\sqrt[3]{x}\) is an integer, then \(x=64\)
- So, the value of x is 64
- Thus, statement 2 alone is also sufficient
Hence the right answer is Option D
20 Sep 2022, 08:00
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Given information: x is an integer; \(2 < x < 100\); \(\sqrt{x}\) also Integer
From this information we can deduce that x is a perfect square so possible values = 4/9/16/25/36/49/64/81
To determine: Value of x
(1) \(\sqrt[3]{x}\) is an integer.
Possible values of x = {4,9,16,25,36,49,64,81}
From these, only 64 satisfies the statement condition. A definite answer
SUFFICIENT
(2) \(2(\sqrt[3]{x}) = \sqrt{x}\)
Possible values of x = {4,9,16,25,36,49,64,81}
\(\sqrt{x}\) = Integer
And if \(2(\sqrt[3]{x}) = \sqrt{x}\) => \((\sqrt[3]{x})\) Also INTEGER
Only 64 satisfies this condition. And also statement condition because Cube Root (64) = 4; 4*2 = 8, which is Square Root (64)
SUFFICIENT
Answer - D
