Number Theory, Division

Division is one of the four basic arithmetic operations, representing the splitting of a quantity into equal parts or finding how many times one number is contained within another. Questions involve performing division, understanding concepts like dividend, divisor, quotient, and remainder, and solving related word problems.

• Question

  In Danny's class there are `30` students. Danny claims that the number of boys is greater by `3` than the number of girls. Is it possible that Danny is right?

  Topics:

  Arithmetic Algebra -> Equations Algebra -> Word Problems Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd)

• Question

  On the circle, there are blue and red points. It is allowed to add a red point and change the colors of its neighboring points or remove a red point and change the colors of its neighboring points (it is not allowed to leave fewer than 2 points on the circle). Prove that it is impossible to move, using only these operations, from a circle with two red points to a circle with two blue points.
  K. Kaznvosky

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Invariants Algebra Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Set Theory Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Colorings -> Chessboard Coloring
  Sources:
  • Tournament of Towns, 1979-1980, Main, Spring Question 1

• Question

  Is it possible to tile a `5xx5` board with dominoes?

  Note: The size of a board square matches the size of a domino square.

  Topics:

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

• Question

  Given two natural numbers `k`  and `m` that differ only in the order of their digits (that is, one is obtained from the other by swapping the order of the digits).

  a. Prove that the sum of the digits of `2k` is equal to the sum of the digits of `2m`.

  b. Prove that if `k` and `m` are even, then the sum of the digits of \(k\over 2\)  is equal to the sum of the digits of \({m \over 2}\).

  c. Prove that the sum of the digits of `5k` is equal to the sum of the digits of `5m`.

  Topics:

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

• 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

• 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