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26 May 2017, 04:42
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We define "A mod n" as the remainder when a positive integer A is divided by n. What is the value of "\(A^{2}\) mod 4"?
(1) A mod 2 = 1
(2) A mod 8 = 1
(1) A mod 2 = 1
(2) A mod 8 = 1
26 May 2017, 04:42
Kudos
Bookmarks
Official Solution:
If you look at the original condition, you only need to know A, so there is 1 variable. In order to match the number of variables to the number of equations, there must be 1 equation. Therefore, D is most likely to be the answer.
In the case of con 1), if A mod 2 = 1, the remainder when A is divided by 2 is 1, so by substituting into \(A = 2P + 1\) (P=any positive integer), you get \(A = 1, 3, 5, 7\ldots\) , and \(A^{2} = 1, 9, 25, 49,..\), so the remainder of everything divided by 4 is 1. That is, \(A^{2}\) mod 4=1, hence it is unique and sufficient.
In the case of con 2), if A mod 8 = 1, the remainder when A is divided by 8 is 1. By substituting into \(A = 8Q + 1\) (Q=any positive integer), \(A =1, 9, 17, 25,\ldots\) , and \(A^{2}=1, 81, 289, 625,..\), so you get the remainder of 1 when everything is divided by 4. That is, \(A^{2}\) mod 4 = 1, hence it is unique and sufficient. The answer is D.
Answer: D
If you look at the original condition, you only need to know A, so there is 1 variable. In order to match the number of variables to the number of equations, there must be 1 equation. Therefore, D is most likely to be the answer.
In the case of con 1), if A mod 2 = 1, the remainder when A is divided by 2 is 1, so by substituting into \(A = 2P + 1\) (P=any positive integer), you get \(A = 1, 3, 5, 7\ldots\) , and \(A^{2} = 1, 9, 25, 49,..\), so the remainder of everything divided by 4 is 1. That is, \(A^{2}\) mod 4=1, hence it is unique and sufficient.
In the case of con 2), if A mod 8 = 1, the remainder when A is divided by 8 is 1. By substituting into \(A = 8Q + 1\) (Q=any positive integer), \(A =1, 9, 17, 25,\ldots\) , and \(A^{2}=1, 81, 289, 625,..\), so you get the remainder of 1 when everything is divided by 4. That is, \(A^{2}\) mod 4 = 1, hence it is unique and sufficient. The answer is D.
Answer: D
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