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Easier than it looks

Sam has a sequence of \(n\) consecutive positive integers \(a_1,a_2,a_3\cdots a_n\) where \(n\) is an odd number and \(a_1\) is an odd number. Adi arranges all of Sam's numbers in some permutation. Prove that given any permutation of the sequence \((p_1,p_2,p_3\cdots p_n)\). The expression:

\[\left(\prod_{k=1}^{\frac {n-1}2}\left(p_{_{2k}}+6^{k}\right)\right)\left(\prod_{k=1}^{\frac{n+1}2} \left(p_{_{2k-1}}+17^{k}\right)\right)\]

will always be divisible by 2.

Details and Assumptions

Basically, in the expression \(p_{a}\) is added to \(6^{\frac{a}2}\) if \(a\) is even and \(17^{\frac{a+1}{2}}\) if \(a\) is odd. Then all the brackets are multiplied together.

Note by Yan Yau Cheng
3 years ago

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