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26 May 2017, 04:42
Kudos
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If m and n are positive integers, what is the remainder when \(11^{mn}\) is divided by 12?
(1) m=even
(2) n=odd
(1) m=even
(2) n=odd
26 May 2017, 04:42
Kudos
Bookmarks
Official Solution:
The 1st of the preliminary knowledge is, in general, for the remainder problem, dividing after the calculation and calculating after the division will both lead to the same value.
The 2nd preliminary knowledge is, when \(11 = 12 * 0 + 11 = 12 * 1 - 1\) is divided by \(12\), \(11\) and \(-1\) have the same values. From \(11 = 12 * 0 + 11 = 12 * 1 - 1\), the number can be divided by \(12\), so that \(11\) and \(-1\) have the same values.
If you solve the original question above, you can take the first step of the variable approach and modify the original condition and the question to get the remainder of \(11^{mn }\) divided by \(12\). According to the 1st preliminary knowledge as mentioned above, you don't calculate \(11^{mn }\) first, but instead, you divide \(11\) first. Then, you get the same values for \(11\) and \(-1\) for dividing the number by \(12\) according to the 2nd preliminary knowledge. So, the remainder of \(11^{mn}\) divided by \(12\) is the same as the remainder of \((-1)^{mn}\) divided by \(12\). If you substitute con 1) m=even=2t (it is any positive integer), you get \((-1) ^{mn}= (-1)^{2tn}= ((-1)^{2}) ^{tn}=1^{tn}=1\), hence it is sufficient. Therefore, the answer is A.
Answer: A
The 1st of the preliminary knowledge is, in general, for the remainder problem, dividing after the calculation and calculating after the division will both lead to the same value.
The 2nd preliminary knowledge is, when \(11 = 12 * 0 + 11 = 12 * 1 - 1\) is divided by \(12\), \(11\) and \(-1\) have the same values. From \(11 = 12 * 0 + 11 = 12 * 1 - 1\), the number can be divided by \(12\), so that \(11\) and \(-1\) have the same values.
If you solve the original question above, you can take the first step of the variable approach and modify the original condition and the question to get the remainder of \(11^{mn }\) divided by \(12\). According to the 1st preliminary knowledge as mentioned above, you don't calculate \(11^{mn }\) first, but instead, you divide \(11\) first. Then, you get the same values for \(11\) and \(-1\) for dividing the number by \(12\) according to the 2nd preliminary knowledge. So, the remainder of \(11^{mn}\) divided by \(12\) is the same as the remainder of \((-1)^{mn}\) divided by \(12\). If you substitute con 1) m=even=2t (it is any positive integer), you get \((-1) ^{mn}= (-1)^{2tn}= ((-1)^{2}) ^{tn}=1^{tn}=1\), hence it is sufficient. Therefore, the answer is A.
Answer: A
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