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14 Jun 2015, 14:39
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If 123 divided by a positive integer \(x\) leaves a remainder of 3, what is the value of \(x\)
(1) The remainder when 60 is divided by \(x\) is more than or equal to 60.
(2) \(x\) is a multiple of 60
(1) The remainder when 60 is divided by \(x\) is more than or equal to 60.
(2) \(x\) is a multiple of 60
14 Jun 2015, 14:39
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Official Solution:
If 123 divided by a positive integer \(x\) leaves a remainder of 3, what is the value of \(x\)
123 divided by a positive integer \(x\) and leaving a remainder of 3 can be expressed as \(123 = xq + 3\), where \(q\) is the quotient. Solving for \(q\), we get \(q = \frac{120}{x}\). This means \(x\) must be a factor of 120 greater than 3 (as the divisor, which is \(x\), must be greater than the remainder, which is 3). Thus, the possible values for \(x\) are 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, or 120.
(1) The remainder when 60 is divided by \(x\) is more than or equal to 60.
Since the remainder cannot be greater than the dividend, this implies that the remainder when 60, the dividend, is divided by \(x\) is 60. Consequently, 60 divided by \(x\) leaving the remainder of 60, implies that \(x\) must be greater than 60. Therefore, \(x = 120\). This statement is sufficient.
(2) \(x\) is a multiple of 60.
From the possible values obtained earlier, \(x\) can be either 60 or 120. . Not sufficient.
Answer: A
If 123 divided by a positive integer \(x\) leaves a remainder of 3, what is the value of \(x\)
123 divided by a positive integer \(x\) and leaving a remainder of 3 can be expressed as \(123 = xq + 3\), where \(q\) is the quotient. Solving for \(q\), we get \(q = \frac{120}{x}\). This means \(x\) must be a factor of 120 greater than 3 (as the divisor, which is \(x\), must be greater than the remainder, which is 3). Thus, the possible values for \(x\) are 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, or 120.
(1) The remainder when 60 is divided by \(x\) is more than or equal to 60.
Since the remainder cannot be greater than the dividend, this implies that the remainder when 60, the dividend, is divided by \(x\) is 60. Consequently, 60 divided by \(x\) leaving the remainder of 60, implies that \(x\) must be greater than 60. Therefore, \(x = 120\). This statement is sufficient.
(2) \(x\) is a multiple of 60.
From the possible values obtained earlier, \(x\) can be either 60 or 120. . Not sufficient.
Answer: A
General Discussion
