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When positive integer \(x\) is divided by 11, the remainder is \(m\) and when positive integer \(y\) is divided by 11, the remainder is \(n\). What is the value of \(m + n\)?
(1) \(x + y\) is a multiple of 11.
(2) \(x - y\) is a multiple of 11.
(1) \(x + y\) is a multiple of 11.
(2) \(x - y\) is a multiple of 11.
31 Dec 2023, 00:47
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Official Solution:
When positive integer \(x\) is divided by 11, the remainder is \(m\) and when positive integer \(y\) is divided by 11, the remainder is \(n\). What is the value of \(m + n\)?
(1) \(x + y\) is a multiple of 11.
If \(x = 1\) and \(y = 10\), then \(m = 1\) and \(n = 10\), giving \(m + n = 11\). However, if both \(x\) and \(y\) are multiples of 11, then \(m = n = 0\), resulting in \(m + n = 0\). Not sufficient. (2) \(x - y\) is a multiple of 11.
If \(x = 12\) and \(y = 1\), then \(m = 1\) and \(n = 1\), giving \(m + n = 2\). However, if both \(x\) and \(y\) are multiples of 11, then \(m = n = 0\), resulting in \(m + n = 0\). Not sufficient.
(1)+(2) Summing the equations from the statements above, we have:
\((x + y) + (x - y) = \text{(a multiple of 11)} + \text{(a multiple of 11)}\)
\(2x = \text{(a multiple of 11)}\)
Consequently, \(x\) must be a multiple of 11. Since \(x + y\) is also a multiple of 11, \(y\) must be a multiple of 11 as well. Therefore, both \(m\) and \(n\) are 0, making \(m + n = 0\). Sufficient.
Answer: C
When positive integer \(x\) is divided by 11, the remainder is \(m\) and when positive integer \(y\) is divided by 11, the remainder is \(n\). What is the value of \(m + n\)?
(1) \(x + y\) is a multiple of 11.
If \(x = 1\) and \(y = 10\), then \(m = 1\) and \(n = 10\), giving \(m + n = 11\). However, if both \(x\) and \(y\) are multiples of 11, then \(m = n = 0\), resulting in \(m + n = 0\). Not sufficient. (2) \(x - y\) is a multiple of 11.
If \(x = 12\) and \(y = 1\), then \(m = 1\) and \(n = 1\), giving \(m + n = 2\). However, if both \(x\) and \(y\) are multiples of 11, then \(m = n = 0\), resulting in \(m + n = 0\). Not sufficient.
(1)+(2) Summing the equations from the statements above, we have:
\((x + y) + (x - y) = \text{(a multiple of 11)} + \text{(a multiple of 11)}\)
\(2x = \text{(a multiple of 11)}\)
Consequently, \(x\) must be a multiple of 11. Since \(x + y\) is also a multiple of 11, \(y\) must be a multiple of 11 as well. Therefore, both \(m\) and \(n\) are 0, making \(m + n = 0\). Sufficient.
Answer: C
