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Is a positive integer \(x\) odd?
(1) The greatest common factor of the smallest positive factor of \(x\) and the largest positive factor of \(x\) is odd
(2) The least common multiple of the smallest positive factor of \(x\) and the largest positive factor of \(x\) is even
(1) The greatest common factor of the smallest positive factor of \(x\) and the largest positive factor of \(x\) is odd
(2) The least common multiple of the smallest positive factor of \(x\) and the largest positive factor of \(x\) is even
11 Oct 2021, 08:06
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Official Solution:
Is a positive integer \(x\) odd?
(1) The greatest common factor of the smallest positive factor of \(x\) and the largest positive factor of \(x\) is odd
The smallest positive factor of any positive integer is 1 and the largest positive factor of any positive integer is the integer itself. So, we are given that the greatest common factor of 1 and \(x\) is odd. This is true for any positive integer \(x\) because the greatest common factor of 1 and \(x\) is always 1, which is odd. So, \(x\) could be any positive even or any positive odd number. Not sufficient.
(2) The least common multiple of the smallest positive factor of \(x\) and the largest positive factor of \(x\) is even
The smallest positive factor of any positive integer is 1 and the largest positive factor of any positive integer is the integer itself. So, we are given that the least common multiple 1 and \(x\) is even. Since the least common multiple 1 and \(x\) is \(x\) itself, then we are basically told that \(x\) is even. Sufficient.
Answer: B
Is a positive integer \(x\) odd?
(1) The greatest common factor of the smallest positive factor of \(x\) and the largest positive factor of \(x\) is odd
The smallest positive factor of any positive integer is 1 and the largest positive factor of any positive integer is the integer itself. So, we are given that the greatest common factor of 1 and \(x\) is odd. This is true for any positive integer \(x\) because the greatest common factor of 1 and \(x\) is always 1, which is odd. So, \(x\) could be any positive even or any positive odd number. Not sufficient.
(2) The least common multiple of the smallest positive factor of \(x\) and the largest positive factor of \(x\) is even
The smallest positive factor of any positive integer is 1 and the largest positive factor of any positive integer is the integer itself. So, we are given that the least common multiple 1 and \(x\) is even. Since the least common multiple 1 and \(x\) is \(x\) itself, then we are basically told that \(x\) is even. Sufficient.
Answer: B
