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Is \(x\) the square of an integer?
(1) When \(x\) is divided by 12, the remainder is 6.
(2) When \(x\) is divided by 14, the remainder is 2.
(1) When \(x\) is divided by 12, the remainder is 6.
(2) When \(x\) is divided by 14, the remainder is 2.
16 Sep 2014, 01:44
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Official Solution:
Is \(x\) the square of an integer?
This question is asking if \(x\) is a perfect square. A perfect square is an integer that can be expressed as the square of an integer. For example, \(16 = 4^2\) is a perfect square.
Remember that a perfect square always has even powers of its prime factors. And it goes both ways: if a number has even powers for all its prime factors, then it's a perfect square. For example: \(36 = 2^2*3^2\), where the powers of the prime factors 2 and 3 are both even.
(1) When \(x\) is divided by 12, the remainder is 6.
The above can be written as \(x = 12q + 6 = 6(2q + 1) = 2*3*(2q + 1)\). Since \(2q + 1\) is an odd number, the power of 2 in \(x\) will be 1, which is odd: \(x = 2*3*odd\). This means \(x\) can't be a perfect square. Sufficient.
(2) When \(x\) is divided by 14, the remainder is 2.
The above can be written as \(x = 14p + 2\). Hence, \(x\) could be 2, 16, 30, etc. This implies \(x\) may or may not be a perfect square. Not sufficient.
Answer: A
Is \(x\) the square of an integer?
This question is asking if \(x\) is a perfect square. A perfect square is an integer that can be expressed as the square of an integer. For example, \(16 = 4^2\) is a perfect square.
Remember that a perfect square always has even powers of its prime factors. And it goes both ways: if a number has even powers for all its prime factors, then it's a perfect square. For example: \(36 = 2^2*3^2\), where the powers of the prime factors 2 and 3 are both even.
(1) When \(x\) is divided by 12, the remainder is 6.
The above can be written as \(x = 12q + 6 = 6(2q + 1) = 2*3*(2q + 1)\). Since \(2q + 1\) is an odd number, the power of 2 in \(x\) will be 1, which is odd: \(x = 2*3*odd\). This means \(x\) can't be a perfect square. Sufficient.
(2) When \(x\) is divided by 14, the remainder is 2.
The above can be written as \(x = 14p + 2\). Hence, \(x\) could be 2, 16, 30, etc. This implies \(x\) may or may not be a perfect square. Not sufficient.
Answer: A
General Discussion
