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

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Try this beautiful Digits Problem from Number theorm from PRMO 2018, Question 19.

Let $N=6+66+666+\ldots \ldots+666 \ldots .66,$ where there are hundred 6 's in the last term in the sum. How many times does the digit 7 occur in the number $N ?$

- $30$
- $33$
- $36$
- $39$
- $42$

Number theorm

Digits Problem

integer

But try the problem first...

Answer:$33$

Source

Suggested Reading

PRMO-2018, Problem 19

Pre College Mathematics

First hint

Given that $\mathrm{N}=6+66+666+....... \underbrace{6666 .....66}_{100 \text { times }}$

If you notice then we can see there are so many large terms. but we have to find out the sum of the digits. but since the number of digits are large so we can not calculate so eassily . we have to find out a symmetry or arrange the number so that we can use any formula taht we can calculate so eassily. if we multiply \(\frac{6}{9}\) then it becomes $=\frac{6}{9}[9+99+\ldots \ldots \ldots \ldots+\underbrace{999 \ldots \ldots \ldots .99}_{100 \text { times }}]$

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

Second Hint

$\mathrm{N}=\frac{6}{9}[9+99+\ldots \ldots \ldots \ldots+\underbrace{999 \ldots \ldots \ldots .99}_{100 \text { times }}]$

$=\frac{6}{9}\left[(10-1)+\left(10^{2}-1\right)+.......+\left(10^{100}-1\right)\right]$

$=\frac{6}{9}\left[\left(10+10^{2}+.....+10^{100}\right)-100\right]$

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

Final Step

$=\frac{6}{9}\left[\left(10^{2}+10^{3}+\ldots \ldots \ldots+10^{100}\right)-90\right]$

$=\frac{6}{9}\left(10^{2} \frac{\left(10^{99}-1\right)}{9}\right)-60$

$=\frac{200}{27}\left(10^{99}-1\right)-60$

$=\frac{200}{27}\underbrace{(999....99)}_{99 \text{times}}-60$

$=\frac{1}{3}\underbrace{(222.....200)}_{99 \mathrm{times}}-60$

$=\underbrace{740740 \ldots \ldots .7400-60}_{740 \text { comes } 33 \text { times }}$ $=\underbrace{740740 \ldots \ldots .740}_{32 \text { times }}+340$

$\Rightarrow 7$ comes 33 times

- 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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