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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, 10 months ago

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