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

  Can you fill a table of size `5xx5` with

  a. Integers,

  b. Real numbers,

   such that the sum of each row is even, and the sum of each column is odd?

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  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Double Counting Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction

• Question

  In two classes with an equal number of students, a quiz was administered. After grading the quiz, the teacher claimed that the number of grades of `0 ` was `13` greater than the number of all other grades combined. Is it possible that he was mistaken?

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  Topics:

  Combinatorics -> Double Counting Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction
  Sources:
  • problems.ru web site

• The Knight and the Dragon

  A knight encountered a dragon with three heads on his way and they began to fight. Every time the knight chops off one of the dragon's heads, three new heads appear in its place. Is it possible that at the end of the battle, the dragon will have a thousand heads?

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  Topics:

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

• Question

  Is it possible to make change for a `25` lira note using `10` coins worth `1`, `3`, and `5` lira?

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  Topics:

  Arithmetic Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd)
  Sources:
  • problems.ru web site

• Question

  Find five natural numbers whose sum is `20`, and whose product is `420`.

  Topics:

  Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample Number Theory -> Prime Numbers -> Prime Factorization
  Sources:
  • problems.ru web site

• Ali Baba and the Forty Thieves

  Ali Baba wrote the number `17` on a piece of paper. The forty thieves pass the paper to each other, and each one either adds `1` to the existing number, or subtracts `1`, until each of them has done so once, and then they return the paper to Ali Baba.

  Is it possible that the number now written on the paper is `40`?

  Topics:

  Arithmetic Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  A knight exited the square `a1` and, after several moves, returned to the same square.

  Is it possible that the knight made an odd number of moves?

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  Topics:

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

• Question

  A knight moves from square `a1` to square `h8`. Is it possible that along the way it visited every square on the board exactly once?

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  Topics:

  Combinatorics -> Double Counting Combinatorics -> Invariants Combinatorics -> Graph Theory Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction Combinatorics -> Colorings -> Chessboard Coloring

• Question

  Every person who ever lived on Earth performed a certain number of handshakes (including 0). Prove that the number of people who performed an odd number of handshakes is even.

  Topics:

  Combinatorics -> Double Counting Combinatorics -> Graph Theory Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction

• Question

  The magical land consists of `25` provinces. Is it possible that each province borders an odd number of other provinces?

  Topics:

  Combinatorics -> Double Counting Combinatorics -> Graph Theory Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction Combinatorics -> Colorings