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16 Sep 2014, 00:39
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If \(x\) and \(y\) are positive integers, is \(\frac{10^x + y}{3}\) an integer?
(1) \(x\) is a multiple of 3
(2) \(y\) is a multiple of 3
(1) \(x\) is a multiple of 3
(2) \(y\) is a multiple of 3
16 Sep 2014, 00:39
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Official Solution:
If \(x\) and \(y\) are positive integers, is \(\frac{10^x + y}{3}\) an integer?
For \(\frac{10^x + y}{3}\) to be an integer, \(10^x + y\) must be divisible by 3. A number is divisible by 3 if the sum of its digits is divisible by 3. Since \(x\) is an integer, the sum of the digits of \(10^x\) is always 1, regardless of the value of \(x\). Thus, we only need additional information about \(y\) to answer the question.
(1) \(x\) is a multiple of 3.
Not sufficient.
(2) \(y\) is a multiple of 3.
Given that \(y\) is a multiple of 3, the sum of the digits of \(10^x + y\) will be 1 + (a multiple of 3), which is one more than a multiple of 3. This implies that \(10^x + y\) is not divisible by 3. Sufficient.
Answer: B
If \(x\) and \(y\) are positive integers, is \(\frac{10^x + y}{3}\) an integer?
For \(\frac{10^x + y}{3}\) to be an integer, \(10^x + y\) must be divisible by 3. A number is divisible by 3 if the sum of its digits is divisible by 3. Since \(x\) is an integer, the sum of the digits of \(10^x\) is always 1, regardless of the value of \(x\). Thus, we only need additional information about \(y\) to answer the question.
(1) \(x\) is a multiple of 3.
Not sufficient.
(2) \(y\) is a multiple of 3.
Given that \(y\) is a multiple of 3, the sum of the digits of \(10^x + y\) will be 1 + (a multiple of 3), which is one more than a multiple of 3. This implies that \(10^x + y\) is not divisible by 3. Sufficient.
Answer: B
11 Oct 2023, 01:16
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
