Question

A positive integer k is said to be good if there exists a partition of {1, 2, 3, ..., 20} into disjoint proper subsets such that the sum of the numbers in each subset of the partition is k. How many good numbers are there?

Answer 1

Possible sum of partition of set {1, 2, 3,.... 20} are

(1) 105, 105

{1, 2, 3, 4, 5, 6, .... 14}, {15, 16, 17, 18, 19, 20}

(2) 70, 70, 70

{20, 19, 15, 16}, {11, 10, 17, 18, 14}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13}

(3) 42, 42, 42, 42, 42

(4) 35, 35, 35, 35, 35, 35

(5) 30, 30, 30, 30, 30, 30, 30

(6) 21, 21, 21, 21, 21, 21, 21, 21, 21, 21