MHB How Can You Prove That 83 Divides x in This Mathematical Series?

[anemone]

Here is this week's POTW:

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Let $x$ and $y$ be the positive integers such that $\dfrac{x}{y}=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\cdots-\dfrac{1}{54}+\dfrac{1}{55}$.

Prove that 83 divides $x$.

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to kaliprasad for his correct solution!(Cool)

You can find the model answer below:

Spoiler

By using the identity $1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\cdots-\dfrac{1}{2n}=\dfrac{1}{n+1}+\dfrac{1}{n+2}+\cdots+\dfrac{1}{2n}$, we have

$\begin{align*}\dfrac{x}{y}&=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\cdots-\dfrac{1}{54}+\dfrac{1}{55}\\&=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\cdots-\dfrac{1}{2(27)}+\dfrac{1}{55}\\&=\dfrac{1}{28}+\dfrac{1}{29}+\dfrac{1}{30}+\cdots+\dfrac{1}{54}+\dfrac{1}{55}\\&=\left(\dfrac{1}{28}+\dfrac{1}{55}\right)+\left(\dfrac{1}{29}+\dfrac{1}{54}\right)+\cdots+\left(\dfrac{1}{41}+\dfrac{1}{42}\right)\\&=\dfrac{83k}{y}\,\,\,\text{where}\,\,\,(83,\,y)=1\end{align*}$

and this completes the proof.