Some questions of Number Theory

I wish to know how to solve these problems. Any help will be appreciated!

Question 1

Find the total number of natural numbers \(n\) for which \(111\) divides \(16^n-1\) where \(n\) is less than \(1000\).

Answer: 111111

Question 2

Find the remainder when (1!)2+(2!)2+(3!)2(100!)2(1!)^2+(2!)^2+(3!)^2 \dots (100!)^2 is divided by 11521152.

Answer: 4141

Question 3

Find the remainder when 321+921+2721+81213^{21}+9^{21}+27^{21}+81^{21} is divided by 320+13^{20}+1.

Answer: 6060

I have also provided the answers, but I wish to know the proper method to solve this, and if there is some trick to solve these questions in general, any opinions are welcome!

Note by Vinayak Srivastava
3 months ago

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1 vote

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You might need to learn Modular arithmetic which is the basic requirement for such questions. Then explore number theory practice on brilliant.

  • Modular Arithmetic
  • Parity of Integers
  • Number Theory
  • Chinese Remainder Theorem
  • Linear Diophantine Equations
  • Fermat's Little Theorem
  • Lucas' Theorem
  • Sum of Squares Theorems
  • Euler's Totient Function
  • Euler's Theorem
  • Finding the Last Digit of a Power
  • Order of an Element
  • Primitive Roots
  • Lifting The Exponent

Mahdi Raza - 3 months ago

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The answer to these specific questions can be found on the internet as well

Mahdi Raza - 3 months ago

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Can you tell the solution of second one also? I think I understand the solution of the third problem given on Quora. Thanks for these two anyway!

Vinayak Srivastava - 3 months ago

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Umm.. I wasn't able to find for the second one. I've seen a similar question where the expression is to be divided by 100. And that is easy because after 10!, all the numbers are divisible by 100. And you can do it that way. Do tell if you find an answer to the second question

Mahdi Raza - 3 months ago

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@Mahdi Raza OK, I'll wait for others to see this. Thanks for these two solutions!

Vinayak Srivastava - 3 months ago

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@Mahdi Raza I found it! I just observed that (6!)2(6!)^2 and so on is divisible by 11521152. So I just squared the first five terms on calculator, and got the correct answer! Your comment was a good hint on what to do! Thanks a lot!

Vinayak Srivastava - 3 months ago

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@Vinayak Srivastava Yeah Márton has done that. Great!

Mahdi Raza - 3 months ago

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@Mahdi Raza I did it first :( But good thinking by you both!

Vinayak Srivastava - 3 months ago

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I solved the problems as @Mahdi Raza linked them. The solution for the second question: 1152=27321152=2^7\cdot 3^2. So from (6!)2(6!)^2 each square is divisible by 1152. Therefore the solution is (1+22+62+242+1202)  mod  1152=41(1+2^2+6^2+24^2+120^2)\;mod\;1152=41.

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Nice!

Mahdi Raza - 3 months ago

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