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.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

There are no comments in this discussion.