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

  Prove that the sum of two consecutive numbers is always odd.

  Topics:

  Number Theory -> Division -> Parity (Even/Odd) Proof and Example

• Question

  Along the street are located `6` trees. One day, `6` parrots arrived and sat on the trees, one parrot on each tree. From time to time, two parrots each move to a neighboring tree of their choice. Can the parrots all gather on the same tree?

  Topics:

  Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd)

• Quiz

  In a class of 25 students, a quiz was given consisting of 7 questions. Prove that at least one of the following two statements is true:

  1. There is a student who solved an odd number of questions.
  2. There is a question that was solved by an even number of students.
  Topics:

  Combinatorics -> Pigeonhole Principle Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 5

• Circle of Liars - The Truth Claim

  In a circle, `n` people are seated, each of whom is either a liar or a truth-teller.

  The people are looking towards the center of the circle. A liar always lies, and a truth-teller always tells the truth.

  Each of the people knows exactly who is a liar and who is a truth-teller.

  Each of the people says that the person sitting two places to their left (that is, next to the person sitting next to them), is a truth-teller.

  It is known that in the circle there is at least one liar, and at least one truth-teller.

  a. Is it possible that `n = 2017`?

  b. Is it possible that `n = 5778`?

  (Solution format: "word, word" for example "cat, puppy")

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  Topics:

  Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction Logic -> Truth-tellers and Liars Problems
  Sources:
  • Gillis Mathematical Olympiad, 2017-2018 Question 1

• Four Numbers

  Given four distinct positive integers. The sum of these numbers is equal to 18.
  Additionally, it is known that the product of these four numbers is odd. Calculate this product.

  Topics:

  Arithmetic Number Theory -> Division -> Parity (Even/Odd)
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 3-4 Question 4

• Five Numbers

  Given five distinct positive integers. The sum of these numbers is 27. Additionally, it is known that the product of these five numbers is odd. Calculate this product.

  Topics:

  Arithmetic Number Theory -> Division -> Parity (Even/Odd)
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 4

• Integer Coefficients?

  Given real numbers a, b, c such that for every integer x, the number `ax^2+bx+c` is an integer. Does this necessarily imply that a, b, c are all integers? Prove it, or provide a counterexample.

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  Topics:

  Algebra Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • Grossman Math Olympiad, 2017, Juniors Question 2

• Equality in Stages

  The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are written on the board, and David is supposed to change them in stages. At each stage, David is allowed to choose two numbers and change them by 1, that is, to add 1 to both, subtract 1 from both, or add 1 to one and subtract 1 from the other.

  Can David, after a number of stages, reach a situation where all the numbers on the board are equal? If so, show an example, and if not, explain your answer in detail.

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  Topics:

  Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Grossman Math Olympiad, 2017, Juniors Question 3

• Two Argue, the Third Takes

  The entire class is in dispute!

  42 think yes, 43 think maybe, and 36 think no.

  When two people who think differently from each other meet, they both change their position to the third.

  What is the minimum number of meetings that must take place until everyone agrees on the same position?

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  Topics:

  Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 9

• Question

  There are 5778 extinguished lamps arranged at equal distances on a circle. Below each lamp is a button. Pressing a button changes the state of 4 lamps: the lamp next to the button, the next two lamps in the circle clockwise, and the lamp opposite the button (an extinguished lamp lights up when its state is changed, and a lit lamp is extinguished). What is the maximum number of lamps that can be lit simultaneously?
  

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Colorings
  Sources:
  • Beno Arbel Olympiad, 2017, Grade 8 Question 8