Let E denote the set of all natural numbers n such that 3 < n < 100 and the set 1,2,3, , n can be partitioned in to 3 subsets with equal sums. Find the number of elements of E .

Step-by-step solution

1,2 . . n This set can be partitioned into 3 subsets with equal sums so total sum is divisible by 3 , n ( n +1) 2 is divisible by 3 . So n will be of the form 3 , 3 +2 or for convenience we can take n =6 , 6 +2,6 +3,6 +5 If n =6 then we can group numbers in bundles of 6 . In each bundle we can select numbers like 1,2,3,4,5,6(16,25,34) If n=6 +2 then we can club last bundle of 8 numbers rest can be partitioned and those 8 numbers can be done 1,2,3,4,5,6,7,8(1236,48) If n =6 +3 , we can club last nine numbers and res