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14 Jun 2015, 14:46
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If both \(x\) and \(y\) are positive integers, what is the remainder when \(2*10^x + 11y\) is divided by 9?
(1) \(x=6\)
(2) \(y=8\)
(1) \(x=6\)
(2) \(y=8\)
14 Jun 2015, 14:46
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Official Solution:
If both \(x\) and \(y\) are positive integers, what is the remainder when \(2*10^x + 11y\) is divided by 9?
For an integer to be divisible by 9, the sum of its digits must be divisible by 9. The sum of the digits of \(2*10^x\) (e.g., 20, 200, 2,000, ...) is always 2, regardless of the value of \(x\). Thus, we only need the value of \(11y\) to determine the divisibility of \(2*10^x + 11y\) by 9.
(1) \(x=6\).
Not sufficient.
(2) \(y=8\).
In this case, \(11y = 88\), so the sum of the digits of \(2*10^x + 11y\) is \(2 + (8 + 8) = 18\), which is a multiple of 9. Therefore, \(2*10^x + 11y\) is divisible by 9. Sufficient.
Answer: B
If both \(x\) and \(y\) are positive integers, what is the remainder when \(2*10^x + 11y\) is divided by 9?
For an integer to be divisible by 9, the sum of its digits must be divisible by 9. The sum of the digits of \(2*10^x\) (e.g., 20, 200, 2,000, ...) is always 2, regardless of the value of \(x\). Thus, we only need the value of \(11y\) to determine the divisibility of \(2*10^x + 11y\) by 9.
(1) \(x=6\).
Not sufficient.
(2) \(y=8\).
In this case, \(11y = 88\), so the sum of the digits of \(2*10^x + 11y\) is \(2 + (8 + 8) = 18\), which is a multiple of 9. Therefore, \(2*10^x + 11y\) is divisible by 9. Sufficient.
Answer: B
15 Sep 2023, 02:34
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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.
