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26 May 2017, 04:42
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If \(a\) and \(b\) are positive integers, is \(a^{4}-b^{4}\) divisible by \(4\)?
(1) \(a+b\) is divisible by \(4\)
(2) The remainder is \(2\) when \(a^{2}+b^{2}\) is divided by \(4\)
(1) \(a+b\) is divisible by \(4\)
(2) The remainder is \(2\) when \(a^{2}+b^{2}\) is divided by \(4\)
26 May 2017, 04:42
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Official Solution:
If you take the 1st step of the variable approach and modify the original condition and the question,
you get \(a^{4}-b^{4}=4t\)? (t=any positive integer), \((a^{2}-b^{2})(a^{2}+b^{2})=4t\)?,
\((a-b)(a+b)(a^{2}+b^{2})=4t\)?. If so, in the case of con 1), since it is \(a+b=4s\) (\(s\)=any positive integer), it becomes \(a^{4}-b^{4}=(a-b)(a+b)(a^{2}+b^{2})=(a-b)4s(a^{2}+b^{2})=4s(a-b)(a^{2}+b^{2})\), hence yes, and sufficient.
If you look at con 2) and apply "CMT 4(B: if you get A or B too easily, consider D)", you get \(a^{2}+b^{2}=4q+2\) (\(q\)=any positive integer) and it is satisfied by \((a,b)=(1,1), (3,3)\), and \((5,1)\). In this case, \(a\)-\(b\) becomes a multiple of 4, hence yes. In the case of \((a,b)=(1,3)\), \(a+b\) becomes a multiple of 4 as well, hence yes. Thus, you always get yes, hence sufficient. Therefore, the answer is D.
Answer: D
If you take the 1st step of the variable approach and modify the original condition and the question,
you get \(a^{4}-b^{4}=4t\)? (t=any positive integer), \((a^{2}-b^{2})(a^{2}+b^{2})=4t\)?,
\((a-b)(a+b)(a^{2}+b^{2})=4t\)?. If so, in the case of con 1), since it is \(a+b=4s\) (\(s\)=any positive integer), it becomes \(a^{4}-b^{4}=(a-b)(a+b)(a^{2}+b^{2})=(a-b)4s(a^{2}+b^{2})=4s(a-b)(a^{2}+b^{2})\), hence yes, and sufficient.
If you look at con 2) and apply "CMT 4(B: if you get A or B too easily, consider D)", you get \(a^{2}+b^{2}=4q+2\) (\(q\)=any positive integer) and it is satisfied by \((a,b)=(1,1), (3,3)\), and \((5,1)\). In this case, \(a\)-\(b\) becomes a multiple of 4, hence yes. In the case of \((a,b)=(1,3)\), \(a+b\) becomes a multiple of 4 as well, hence yes. Thus, you always get yes, hence sufficient. Therefore, the answer is D.
Answer: D
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