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16 Nov 2021, 09:24
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Are positive integers \(m\) and \(n\) both multiples of 3?
(1) The two-digit number \(mn\), where \(m\) represents the tens digit and \(n\) represents the units digit, is a multiple of 9.
(2) \(n\) is a multiple of 3.
(1) The two-digit number \(mn\), where \(m\) represents the tens digit and \(n\) represents the units digit, is a multiple of 9.
(2) \(n\) is a multiple of 3.
16 Nov 2021, 09:24
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Official Solution:
Are positive integers \(m\) and \(n\) both multiples of 3?
(1) The two-digit number \(mn\), where \(m\) represents the tens digit and \(n\) represents the units digit, is a multiple of 9.
An integer is a multiple of 9 if the sum of its digit is a multiple of 9. So, \(m+n\) must be a multiple of 9.
If two-digit number \(mn\) is 36, 63 or 99, then the answer is YES;
If two-digit number \(mn\) is 18, 27, 45, 54, 72 or 81, then the answer is NO.
(2) \(n\) is a multiple of 3
No info about \(m\). Not sufficient.
(1)+(2) If \(m\) is not a multiple of 3, then \(m + n\) would be the sum of a non-multiple of 3 and a multiple of 3, resulting in a non-multiple of 3. However, we know that \(m + n\) is a multiple of 9, so it must also be a multiple of 3. Thus, \(m\) must be a multiple of 3 as well. Sufficient.
Answer: C
Are positive integers \(m\) and \(n\) both multiples of 3?
(1) The two-digit number \(mn\), where \(m\) represents the tens digit and \(n\) represents the units digit, is a multiple of 9.
An integer is a multiple of 9 if the sum of its digit is a multiple of 9. So, \(m+n\) must be a multiple of 9.
If two-digit number \(mn\) is 36, 63 or 99, then the answer is YES;
If two-digit number \(mn\) is 18, 27, 45, 54, 72 or 81, then the answer is NO.
(2) \(n\) is a multiple of 3
No info about \(m\). Not sufficient.
(1)+(2) If \(m\) is not a multiple of 3, then \(m + n\) would be the sum of a non-multiple of 3 and a multiple of 3, resulting in a non-multiple of 3. However, we know that \(m + n\) is a multiple of 9, so it must also be a multiple of 3. Thus, \(m\) must be a multiple of 3 as well. Sufficient.
Answer: C
25 May 2024, 10:54
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