Problem on Largest Prime Factor | PRMO 2019 | Question 21

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

Answer: is 17.

PRMO, 2019, Question 21

Elementary Number Theory by David Burton

Try with Hints

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

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

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.

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.