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

  The number `458` is written on the board. In each single step, you are allowed to either multiply the number written on the board by `2`, or erase its last digit.

  Is it possible to obtain the number `14` using these operations?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  Is it possible for the sum of three natural numbers to be divisible by each of them?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • problems.ru web site

• Question

  Prove that the difference of squares of two consecutive odd numbers is divisible by `8`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities
  Sources:
  • problems.ru web site

• Question

  Prove that the product of three consecutive numbers is divisible by `6`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Number Theory -> Prime Numbers -> Prime Factorization

• Question

  Prove that the product of four consecutive numbers is divisible by `24`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Prime Numbers -> Prime Factorization

• Question

  Prove that for every prime number `p>3 ` the following holds: `p^2-1` is divisible by `6`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Number Theory -> Division -> Parity (Even/Odd) Proof and Example Number Theory -> Prime Numbers -> Prime Factorization

• Question

  `a,b` are two distinct natural numbers. The sum of the divisors of each is equal to the same natural number `n`. What is the smallest possible value of `n`?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Prime Numbers -> Prime Factorization
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 1

• Question

  A. You have a large jug of 12 liters of olive oil and two empty smaller vessels, one of 5 liters and one of 8 liters. Can you divide the oil you have into two equal parts, if you only have these vessels and no additional measuring tools?

  B. The same question, but instead of the 5-liter vessel, you have a 4-liter vessel.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) -> Euclidean Algorithm Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction

• Question

  In the magical land, there are only coins of `5`, `6`, and `15` liras. Shlomi currently only has coins of `6` and `15` liras. Shlomi wants to buy a book that costs `38` liras. Will he be able to pay for the book without change?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division

• Baobab

  In the following exercise, identical digits have been replaced with identical letters, and different digits have been replaced with different letters. Reconstruct the exercise.

  `BAOxxBAxxB = 2002`

   

   

   

   

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Number Theory -> Prime Numbers -> Prime Factorization Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic