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

  Prove that the product of four consecutive numbers is divisible by `24`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Prime Numbers -> Prime Factorization

• Question

  Prove that for every prime number `p>3 ` the following holds: `p^2-1` is divisible by `6`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Number Theory -> Division -> Parity (Even/Odd) Proof and Example Number Theory -> Prime Numbers -> Prime Factorization

• Question

  `a,b` are two distinct natural numbers. The sum of the divisors of each is equal to the same natural number `n`. What is the smallest possible value of `n`?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Prime Numbers -> Prime Factorization
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 1

• Question

  A. You have a large jug of 12 liters of olive oil and two empty smaller vessels, one of 5 liters and one of 8 liters. Can you divide the oil you have into two equal parts, if you only have these vessels and no additional measuring tools?

  B. The same question, but instead of the 5-liter vessel, you have a 4-liter vessel.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) -> Euclidean Algorithm Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction

• Question

  In the magical land, there are only coins of `5`, `6`, and `15` liras. Shlomi currently only has coins of `6` and `15` liras. Shlomi wants to buy a book that costs `38` liras. Will he be able to pay for the book without change?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division

• 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

• Baobab

  In the following exercise, identical digits have been replaced with identical letters, and different digits have been replaced with different letters. Reconstruct the exercise.

  `BAOxxBAxxB = 2002`

   

   

   

   

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Number Theory -> Prime Numbers -> Prime Factorization Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic

• Question

  Find all numbers that are divisible by 30 and have exactly 30 distinct divisors (enter the number of such numbers to check your answer)

  Show hint

  Log in here to receive the hint
  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Prime Numbers -> Prime Factorization
  Sources:
  • Tournament of Towns, 1981-1982, Spring

• Question

  Given the sequence `1 , 1/2 ,1/3 ,1/4 ,1/5,...`, does there exist an arithmetic sequence composed of terms from the aforementioned sequence?

  1. Of length 5

  2. Of any length

  Show hint

  Log in here to receive the hint
  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Arithmetic -> Fractions Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM)
  Sources:
  • Tournament of Towns, 1981-1982, Spring

• 5 but not 7

  How many numbers from 1 to 100 (inclusive) are divisible by 5 but not divisible by 7?
   

  Show hint

  Log in here to receive the hint
  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 6