Number Theory

Number Theory is a branch of mathematics concerned with the properties of integers. Topics include prime numbers, divisibility, congruences (modular arithmetic), Diophantine equations, and functions of integers. Questions often require analytical and creative thinking about numbers.

• Question

  Take any three-digit number and arbitrarily rearrange its digits. Prove that the difference between the original number and the new number is divisible by `3`. Is this also true for four-digit numbers?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9
  Sources:
  • problems.ru web site

• Question

  Does there exist a natural number which, when divided by the sum of its digits with a remainder, yields `2017` as both the quotient and the remainder?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Proof and Example -> Proof by Contradiction

• Question

  Prove that the sum of the digits of a perfect square cannot be equal to `2019 `.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Proof and Example -> Proof by Contradiction

• Question

  Given two numbers such that one is obtained from the other by changing the order of its digits. Prove that their difference is divisible by `9`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Proof and Example Arithmetic -> Division with Remainder

• Question

  Given a natural number `n` that is three times greater than the sum of its digits. Prove that `n` is divisible by `27`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Proof and Example

• Question

  Let `n` be a natural number. Prove that if the sum of the digits of `5n` is equal to the sum of the digits of `n`, then `n` is divisible by `9`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9

• Palindromic Number

  Find a four-digit palindromic number that is divisible by 25 and not divisible by 3.

  Note: A palindromic number is a number that does not change if its digits are read in reverse order. For example, the number 5775 is a palindromic number, and the number 5778 is not a palindromic number.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Division -> Parity (Even/Odd) Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 5 and 25 Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 5

• 50 to the Power of

  Show that in the rightmost 504 digits of `1+50+50^2+...+50^1000`

  Each digit appears a number of times divisible by 12

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Algebraic Techniques -> Telescoping Sums
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 2

• Divisible by 2 or 5 but not 3

  How many five-digit numbers are divisible by 2 or 5, but not divisible by 3?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Combinatorics -> Product Rule / Rule of Product
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 4

• 6 on the Board

  The number 6 is written on the board. At each step, you can add the digit 6 to the end of the number (so that it is the units digit,) or replace the number with the sum of its digits.
  Which numbers can be obtained in this way? Describe the entire set of numbers and explain why there are no more

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Combinatorics -> Induction (Mathematical Induction) Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 3-4 Question 6
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 5