What is the NO-SHORTCUT approach for learning great Mathematics?

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Try this beautiful good numbers problem from Number theory from 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?

- $4$
- $6$
- $8$
- $10$
- $2$

Number theorm

good numbers

subset

But try the problem first...

Answer:$6$

Source

Suggested Reading

PRMO-2018, Problem 22

Pre College Mathematics

First hint

**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 ..........

Second Hint

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........

Final Step

**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 $6$

- https://www.cheenta.com/ordered-pairs-prmo-2019-problem-18/
- https://www.youtube.com/watch?v=h_x9kS-J1XY

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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