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.

Prime Numbers Chinese Remainder Theorem Modular Arithmetic / Remainder Arithmetic Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Triangular Numbers Division

• Question

  All numbers from `45` to `65` are written on a board. How many times does the digit `5` appear?

  Topics:

  Number Theory Arithmetic
  Sources:
  • problems.ru web site

• Question

  On Alice's birthday, her friends are asked how old she is. The Mad Hatter says that Alice's age is greater than `11`, and the Cheshire Cat says that her age is greater than `10`. It is known that exactly one of them is wrong. How old is Alice now? Explain!

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  Topics:

  Number Theory Algebra -> Inequalities Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  Is it possible to arrange all the numbers from `1` to `100` in a row such that the difference between any two adjacent numbers is at least `50`? If so, provide an example; if not, prove why.

  Topics:

  Number Theory Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • problems.ru web site

• Question

  The sum of six distinct natural numbers is equal to `22`. Find these numbers.

  Topics:

  Number Theory Arithmetic
  Sources:
  • problems.ru web site

• Question

  Find a two-digit number that is twice as large as the product of its digits.

  Topics:

  Number Theory Arithmetic Algebra -> Algebraic Techniques Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Algebra -> Equations -> Diophantine Equations
  Sources:
  • problems.ru web site

• Question

  Prove that any real number can be written as the sum of 9 numbers, each of which is composed only of the digits 0 and 7.

  Topics:

  Number Theory Arithmetic Proof and Example Algebra -> Sequences
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 3

• Question

  Find all natural numbers with the following property: when divided by 7, their remainder is equal to their quotient.

  Topics:

  Number Theory Algebra -> Inequalities Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Arithmetic -> Division with Remainder

• Question

  Find all two-digit numbers `A` such that the square of the sum of its digits is equal to the sum of the digits of `A^2`.

  Topics:

  Number Theory Arithmetic Algebra -> Inequalities Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  Can the product of two consecutive natural numbers be equal to the product of two consecutive even numbers?

  Topics:

  Number Theory Algebra -> Inequalities Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction

• Question

  Given a real number `a` such that `a+1/a` is an integer. Prove that `a^n+1/a^n` is also an integer for every natural number `n`.

  Topics:

  Number Theory Combinatorics -> Induction (Mathematical Induction) Algebra -> Algebraic Techniques Proof and Example Algebra -> Sequences