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Try this beautiful problem from Pre RMO, 2019 based on the Number theory.

## Number Theory – PRMO 2019

Let \(x_1\) be a positive real number and for every integer n>=1 let \(x_{n+1}=1+x_{1}x_{2}…x_{n-1}x_{n}\). If \(x_{5}=43\). what is the sum of digits of the largest prime factor of \(x_{6}\).

- is 13
- is 25
- is 840
- cannot be determined from the given information

**Key Concepts**

**S**equence

Series

Number System

## Check the Answer

But try the problem first…

Answer: is 13.

Source

Suggested Reading

PRMO, 2019

Elementary Number Theory by David Burton

## Try with Hints

First hint

Here \(x_5=1+x_1x_2x_3x_4\) then \(x_1x_2x_3x_4=42\)

Second Hint

\(x_6=1+x_1x_2x_3x_4x_5\)=1+(42)(43)=1807=(13)(139)

Final Step

Then largest prime factor=139 then sum of digits=13

## Other useful links

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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