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If \(m\) and \(n\) are positive integers, do they yield the same remainder when divided by 24?
(1) \(m\) and \(n\) yield the same remainder when divided by 8.
(2) \(|m - n|\) is a multiple of 9.
(1) \(m\) and \(n\) yield the same remainder when divided by 8.
(2) \(|m - n|\) is a multiple of 9.
12 Nov 2021, 08:04
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Official Solution:
If \(m\) and \(n\) are positive integers, do they yield the same remainder when divided by 24?
If the remainders were the same then we'd be able to write \(m=24p+r\) and \(n=24q+r\).
Subtract one from another \(m-n=24(p-q)\). So, when \(m\) and \(n\) yield the same remainder when divided by 24, their difference is a multiple of 24.
(1) \(m\) and \(n\) yield the same remainder when divided by 8.
The same way as above, we can deduce that this statement implies that \(m-n\) is a multiple of 8. So, \(m-n\) may or may not be a multiple of 24. Not sufficient.
(2) \(|m - n|\) is a multiple of 9.
This means that \(m-n\) is a multiple of 3 but we don't know whether it is also a multiple of 8. Not sufficient.
(1)+(2) From (1) \(m-n\) is a multiple of 8 and from (2) it's is a multiple of 3, thus \(m-n\) is a multiple of 24. Sufficient.
Answer: C
If \(m\) and \(n\) are positive integers, do they yield the same remainder when divided by 24?
If the remainders were the same then we'd be able to write \(m=24p+r\) and \(n=24q+r\).
Subtract one from another \(m-n=24(p-q)\). So, when \(m\) and \(n\) yield the same remainder when divided by 24, their difference is a multiple of 24.
(1) \(m\) and \(n\) yield the same remainder when divided by 8.
The same way as above, we can deduce that this statement implies that \(m-n\) is a multiple of 8. So, \(m-n\) may or may not be a multiple of 24. Not sufficient.
(2) \(|m - n|\) is a multiple of 9.
This means that \(m-n\) is a multiple of 3 but we don't know whether it is also a multiple of 8. Not sufficient.
(1)+(2) From (1) \(m-n\) is a multiple of 8 and from (2) it's is a multiple of 3, thus \(m-n\) is a multiple of 24. Sufficient.
Answer: C
