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Number Theory | PRMO 2019 | Problem 3

Try this beautiful problem from Pre RMO, 2019 based on the number theory. You may use sequential hints to solve the problem.

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

Check the Answer


Answer: is 13.

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

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