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Good numbers Problem | PRMO-2018 | Question 22

Try this beautiful good numbers problem from Number theory from PRMO 2018, Question 22.

Good numbers Problem - PRMO 2018, Question 22


A positive integer $k$ is said to be good if there exists a partition of ${1,2,3, \ldots, 20}$ 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?

  • $4$
  • $6$
  • $8$
  • $10$
  • $2$

Key Concepts


Number theorm

good numbers

subset

Check the Answer


Answer:$6$

PRMO-2018, Problem 22

Pre College Mathematics

Try with Hints


What is good numbers ?

A good number is a number in which every digit is larger than the sum of digits of its right (all less significant bits than it). For example, 732 is a good number, $7>3+2$ and $3>2$ .

Given that $k$ is said to be good if there exists a partition of ${1,2,3, \ldots, 20}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. Now at first we have to find out sum of these integers ${1,2,3, \ldots, 20}$. Later create some partitions such that two partitions be disjoint set and sum of the numbers of these partitions be good numbers

Can you now finish the problem ..........

Sum of numbers equals to $\frac{20 \times 21}{2}=210 \& 210=2 \times 3 \times 5 \times 7$

So $\mathrm{K}$ can be 21,30,35,47,70,105

Can you finish the problem........

Case 1 :

$\mathrm{A}=\{1,2,3,4,5,16,17,18,19,20\}$, $\mathrm{B}=\{6,7,8,9,10,11,12,13,14,15\}$

Case 2 :

$A=\{20,19,18,13\}$, $B=\{17,16,15,12,10\}$, $C=\{1,2,3,4,5,6,7,8,9,11,14\}$

Case 3 :

$\mathrm{A}=\{20,10,12\}$, $\mathrm{B}=\{18,11,13\}$, $\mathrm{C}=\{16,15,9,2\}$, $\mathrm{D}=\{19,8,7,5,3\}$, $\mathrm{E}=\{1,4,6,14,17\}$

Case 4 :

$A=\{20,10\}, B=\{19,11\}$,$ C=\{18,12\}, D=\{17,13\}$,$ E=\{16,14\}$, $F=\{1,15,5\},$
$G=\{2,3,4,6,7,8\}$

Case 5 :

$A=\{20,15\}$, $B=\{19,16\}$, $C=\{18,17\}$, $D=\{14,13,8\}$, $E=\{12,11,10,2\},$
$F=\{1,3,4,5,6,7,9\}$

Case 6 :

$A=\{1,20\}$,$ B=\{2,19\}$, $C=\{3,18\} \ldots \ldots \ldots \ldots$, $J=\{10,11\}$

Therefore Good numbers equal to $6$

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