Question

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

(a) 5 

(b) 6 

(c) 7 

(d) 4

Answer 1

Correct option is (b) 6

Let us partition in to n parts and each part has sum = K, then

nK = 1 + 2 + 3 .... + 20

nK = 210

∴K divides 210. 

Also, K must be ≥ 20

Now, 210 = 2 x 3 x 5 x 7

So, proper divisor of 210 are {1, 2, 3, 5, 6,7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210}

⇒ K can be 21, 30, 35, 42, 70,105

For K = 21, we have (1, 20) (2,19) .....(10, 11)

⇒ 21 is a good number. 

For K = 42, join two-two pairs

For K = 105, join five-five pairs 

⇒ 42 and 105 are also good numbers

For K = 30, we have

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

⇒ K = 30 is also good number 

Similarly, 35 and 70 also good numbers. 

∴There are total 6 good numbers.