Number Theory, Modular Arithmetic / Remainder Arithmetic

Modular Arithmetic (or Remainder Arithmetic) is a system where numbers 'wrap around' after reaching a certain value, the modulus. It deals with congruences and remainders. Questions involve solving equations in modular systems, finding powers modulo `n`, and applications in patterns or cryptography.

• Question

  Given natural numbers n, a, b such that `3n+1=a^2` and `4n+1=b^2`, prove that:

  a. n is divisible by 8 (easier)

  b. n is divisible by 56

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra -> Equations -> Diophantine Equations
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 5

• Question

  Given a regular polygon with n vertices. Calculate the number of distinct (non-congruent) triangles whose vertices coincide with the vertices of the polygon.

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  Topics:

  Combinatorics Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 4 Question 3

• What's the Number?

  Find a three-digit number that, when we add 1 to it, is divisible by 7; when we add 2 to it, is divisible by 8; and when we add 3 to it, is divisible by 9.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 4

• Question

  Consider all the numbers from 1 to 4242. Find the difference between the number of odd numbers divisible by 3
  and the number of numbers divisible by 7 in this range.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Arithmetic -> Fractions Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 5

• Pumbaa and the Candies

  Pumbaa has 11 chocolate candies and 13 toffee candies. Each time he can eat either two candies of different types,
  or three candies of the same type. What is the largest number of candies that Pumbaa can eat according to these rules?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Combinatorics -> Invariants Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 6

• Question

  A two-digit number is written on the board.
  Avi said: "The units digit of the number is 3"
  Beni said: "It's a square number"
  Gili said: "This number is a multiple of 12"
  Then the teacher said: "There are two correct statements and one wrong one here."
  What number was written on the board?

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Logic -> Truth-tellers and Liars Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 1

• Question

  Vered subtracted a number composed of the same digits written in reverse order from a three-digit number.
  As a result, she obtained a two-digit number. Find the two-digit number she obtained.

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra Logic -> Reasoning / Logic
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 2

• 2 or 5 but not 3

  How many two-digit numbers are divisible by 2 or 5, and not divisible by 3?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 6

• Question

  A two-digit number is written on the board.
  Avi said: "The digit 5 appears in this number."
  Beni said: "This is a square number."
  Gili said: "This number is greater than 50."
  Dani said: "The number is divisible by 7."
  Then the teacher said: "There are three correct statements here and one incorrect statement.".
  What number was written on the board?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 1

• Consecutive Numbers

  a. Avi wants to find 10 consecutive numbers whose sum is divisible by 90. Will he succeed?
  b. Benny wants to find 11 consecutive numbers such that their sum is divisible by 90. Will he succeed?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 4