Try this beautiful problem from the Pre-RMO, 2019 based on Largest prime factor.

## Problem on Largest Prime Factor – PRMO 2019

Consider the set E={5,6,7,8,9}, for any partition {A,B} of E, with both A and B non empty. Consider the number obtained by adding the product of elements of A to the product of elements of A to the product of elements of B. Let N be the largest prime number among these numbers, find the sum of the digits of N.

- is 107
- is 17
- is 840
- cannot be determined from the given information

**Key Concepts**

Largest prime

Divisibility

Integer

## Check the Answer

But try the problem first…

Answer: is 17.

PRMO, 2019, Question 21

Elementary Number Theory by David Burton

## Try with Hints

First hint

here one of the set A or set B contains odd number only

Second Hint

set A set B

I 5 6,7,8,9 5+(6)(7)(8)(9)=3029 not prime

II 7 5,6,8,9 7+(5)(6)(7)(8)(9)=2167 not prime

III 9 5,6,7,8 9+(5)(6)(7)(8)=not prime

Final Step

IV 5,7 6,8,9 (5)(7)+(6)(8)(9)=467 prime

V 5,9 6,7,8 (5)(9)+(6)(7)(8)=not prime

VI 7,9 5,6,8 (7)(9)+(5)(6)(8)=not prime

VII 5,7,9 6,8 (5)(7)(9)+(6)(8)not prime

N=467

or, 4+6+7=17.

## Other useful links

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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