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 B . Let N be the largest prime number among these numbers. Find the sum of the digits of N .

Step-by-step solution

one of the set A or set B contain only odd number Set A Set B Comment Case I 5 6,7,8,9 5+6 7 8 9=3029 Which is divisible by 13 II 7 5,6,8,9 7+5 6 8 9=2167 Which is divisible by 11 III 9 5,6,7,8 9+5 6 7 8 is multiple of 3 IV 5,7 6,8,9 5 7+6 8 9=467 which is prime V 5,9 6,7,8 5 9+6 7 8 is divisible by 3 VI 7,9 5,6,8 7 9+5 6 8 is divisible by 3 VII 5,7,9 6,8 5 7 9+6 8 is divisible by 3 Hence N is 467 Sum of the digit of N is 17