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).

• Question

  Does there exist a perfect square whose digits sum to `2001`?

  Justify or provide an example!

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Division -> Parity (Even/Odd) Number Theory -> Prime Numbers -> Prime Factorization Proof and Example -> Proof by Contradiction
  Sources:
  • problems.ru web site

• Question

  It is known that every prime number has two divisors – `1` and the number itself. What numbers have exactly three divisors?

  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization
  Sources:
  • problems.ru web site

• Question

  The number `100` is written on the board. Find a digit that satisfies the following condition:

  If we add it to the notation of the number written on the board once to the left and once to the right, we get a number that is divisible by `12`.

  Show hint

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  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 2, 4, and 8 Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Prime Numbers -> Prime Factorization
  Sources:
  • problems.ru web site

• Question

  Prove that the product of three consecutive numbers is divisible by `6`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Number Theory -> Prime Numbers -> Prime Factorization

• 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

  How many solutions in natural numbers are there to the equation `(2013 - x)(2013-y)=2013^2`?

  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization Algebra -> Equations -> Diophantine Equations
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 6

• Question

  Does there exist a natural number such that the product of its digits is equal to `99`?

  Show hint

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  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization

• 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