Number Theory

Number Theory is a branch of mathematics concerned with the properties of integers. Topics include prime numbers, divisibility, congruences (modular arithmetic), Diophantine equations, and functions of integers. Questions often require analytical and creative thinking about numbers.

• Question

  In the following arithmetic puzzle, different digits have been replaced by different letters, and identical digits – by identical letters. Reconstruct the puzzle:

  `BAOxxBAxxB=2002`

  Topics:

  Arithmetic Algebra -> Equations Algebra -> Inequalities Logic -> Reasoning / Logic Number Theory -> Prime Numbers -> Prime Factorization Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic

• 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`

  Show hint

  Log in here to receive the hint
  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

• Movie Buffs

  Avi, Beni, and Gili love movies. Avi goes to the movies every `3` days, Beni – every `5` days, and Gili goes every `7` days. Today, they all went to the movies together. In how many days can this happen again?

  Topics:

  Arithmetic Algebra -> Word Problems Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM)
  Sources:
  • problems.ru web site

• Question

  The "Sweet Math" candies are sold in boxes of `12` units, and the "Geometry with Nuts" candies – in boxes of `15` units.

  What is the minimum number of boxes that must be purchased so that there are equal quantities of candies of both types?

  Topics:

  Arithmetic Algebra -> Word Problems Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Minimum and Maximum Problems / Optimization Problems
  Sources:
  • problems.ru web site

• THE MARKET WOMEN

  A number of market women sold their various products at a certain price per pound (different in every case), and each received the same amount—`2`s. `2`½d. What is the greatest number of women there could have been? The price per pound in every case must be such as could be paid in current money.
  Topics:

  Arithmetic Algebra -> Word Problems Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 18

• Question

  A vendor has two rolls of fabric, each `36` meters long. He sells the fabric in pieces of `3` meters. How many times will the vendor need to cut the fabric?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Number Theory -> Division
  Sources:
  • problems.ru web site

• Question

  Two workers pave two kilometers of road within two days. How many workers are needed to pave `20` kilometers of road within `20` days?

  Topics:

  Arithmetic Algebra -> Equations Algebra -> Word Problems Logic -> Reasoning / Logic Number Theory -> Division