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

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Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Integers and divisibility.

300 digit number with all digits equal to 1 is

- divisible neither by 37 nor by 101
- divisible by both 37 and 101
- divisible by 37 and not by 101
- divisible by 101 and not by37

Integers

Remainders

Divisibility

But try the problem first...

Answer: divisible by 37 and 101

Source

Suggested Reading

B.Stat Objective Problem 89

Challenges and Thrills of Pre-College Mathematics by University Press

First hint

here we take 300 digit number all digit 1s

Second Hint

111...11=\(\frac{999...99}{9}\)(300 digits)

=\(\frac{10^{300}-1}{9}\)=\(\frac{(10^{3})^{100}-1}{9}\)=\(\frac{(10^{3}-1)X}{9}\)

since \(10^{3}-1\)=999 is divisible by 37 then 111...11(300 digits) is divisible by 37

Final Step

111...11=\(\frac{999...99}{9}\)(300 digits)

=\(\frac{10^{300}-1}{9}\)=\(\frac{(10^{4})^{75}-1}{9}\)=\(\frac{(10^{4}-1)Y}{9}\)

since \(10^{4}-1\)=9999 is divisible by 101 then 111...11(300 digits) is divisible by 101.

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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