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# Number Theory | PRMO 2019 | Problem 3

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 $n\geq1$ 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

Sequence

Series

Number System

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