Problem on Largest Prime Factor | PRMO 2019 | Question 21

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.

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.

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