Number Theory, Prime Numbers

Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. This topic explores their properties, identification, distribution (e.g., Sieve of Eratosthenes), and their fundamental role in number theory (e.g., prime factorization).

• 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)

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  Topics:

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

• Pairwise Relatively Prime Composite Numbers

  Yossi writes two-digit composite numbers on the board. He wants all the numbers written on the board to be pairwise relatively prime.
  What is the maximum number of integers Yossi can write on the board?
  Note: Integers are called relatively prime if they have no common factors other than 1.

  Topics:

  Logic -> Reasoning / Logic Number Theory -> Prime Numbers -> Prime Factorization Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM)
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 5-6 Question 4

• Factoring and Using the Formula

  An interesting formula is `x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)`.

  A: Use it to factor the expression `a^n-b^n`.

  B: Factor the expression `a^n+b^n` for any odd integer n.

  C: Prove that if `2^n-1` is prime, then n is also prime.

  D: Prove that if `2^n+1` is prime, then n is necessarily a power of 2, which is equivalent to `n=2^m`

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  Topics:

  Algebra -> Algebraic Techniques Number Theory -> Prime Numbers -> Prime Factorization Proof and Example -> Proof by Contradiction
  Sources:
  • Junior Science Team Exercises, 2024-2025, Assignment 1 Question 4

• The Number with the Most Divisors

  Among the positive integers less than 1000, which number has the most divisors?

  Topics:

  Arithmetic Number Theory -> Prime Numbers -> Prime Factorization Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Number Theory -> Division
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 3

• THE SULTAN'S ARMY

  A certain Sultan wished to send into battle an army that could be formed into two perfect squares in twelve different ways. What is the smallest number of men of which that army could be composed? To make it clear to the novice, I will explain that if there were `130` men, they could be formed into two squares in only two different ways—`81` and `49`, or `121` and `9`. Of course, all the men must be used on every occasion.
  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 136