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

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Quantrex Academy · Accepted Answer

Correct answer: 6

Sum of numbers equals to (20 × 21)/(2)=210 & 210=2 × 3 × 5 × 7 Number of Partition Sum I 2 105 II 3 70 III 5 42 IV 7 30 V 6 35 VI 10 21 So K can be 21,30,35,47,70,105 Good numbers equal to 6 Case-I: A=\ 1,2,3,4,5,16,17,18,19,20\ , B=\ 6,7,8,9,10,11,12,13,14,15\ Case-II: A=\ 20,19,18,13\ , B=\ 17,16,15,12,10\ , C=\ 1,2,3,4,5,6,7,8,9,11,14\ Case-III: A=\ 20,10,12\ , B=\ 18,11,13\ , C=\ 16,15,9,2\ , D=\ 19,8,7,5,3\ , E=\ 1,4,6,14,17\ Case-IV: aligned A & =\ 20,10\ , B=\ 19,11\ , C=\ 18,12\ , D=\ 17,13\ , E=\ 16,14\ , F=\ 1,15,5\ , G & =\ 2,3,4,6,7,8\ aligned Case-V: aligned A & =\ 20,15\ , B=\ 19,16\ , C=\ 18,17\ , D=\ 14,13,8\ , E=\ 12,11,10,2\ , F & =\ 1,3,4,5,6,7,9\ aligned Case-VI: A=\ 1,20\ , B=\ 2,19\ , C=\ 3,18\ ., J=\ 10,11\

A positive integer k is said to be good if there exists a partition...

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