Number Theory, Division, Parity (Even/Odd)

Parity refers to whether an integer is even (divisible by 2) or odd (not divisible by 2). Many problems, especially in number theory and combinatorics, can be solved or simplified by considering the parity of the numbers involved. Questions often require analyzing how operations affect parity.

• Question

  Does there exist a perfect square that ends with the digits `...2017`?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction

• Question

  `19` apple trees are arranged in a circle. Prove that there exists a pair of adjacent trees such that the total number of apples on them is even.

  Topics:

  Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction

• Question

  In the magical land, there are only coins of `4`, `7`, and `10` liras. Shlomi currently only has coins of `4` and `10` liras. Shlomi wants to buy a book that costs `23` liras. Will he be able to pay for the book without change?

  Topics:

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

• Question

  Can you find two numbers such that both their sum and their product are odd? Justify your answer or provide an example!

  Topics:

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

• Question

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

  Topics:

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

• Question

  Consider the integers from `1` to `700`.

  a. How many of these numbers are even?

  b. How many of these numbers are divisible by `7`?

  c. How many of these numbers are not divisible by `2` nor by `7`?

  Answer question c.

  Topics:

  Number Theory -> Prime Numbers Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Combinatorics -> Matchings Number Theory -> Division -> Parity (Even/Odd) Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Set Theory

• 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)

• Palindromic Number

  Find a four-digit palindromic number that is divisible by 25 and not divisible by 3.

  Note: A palindromic number is a number that does not change if its digits are read in reverse order. For example, the number 5775 is a palindromic number, and the number 5778 is not a palindromic number.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Division -> Parity (Even/Odd) Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 5 and 25 Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 5

• Question

  Is there a solution in natural numbers to the equation `x^2 + 12 = y^3` such that

  a. x is even (easier)

  b. x is odd

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  Topics:

  Number Theory -> Prime Numbers Number Theory -> Modular Arithmetic / Remainder Arithmetic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction Arithmetic -> Division with Remainder Algebra -> Equations -> Diophantine Equations
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 6

• 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