Number Theory, Prime Numbers, Prime Factorization

Prime Factorization is the process of expressing a composite number as a unique product of prime numbers. Questions involve finding the prime factorization of integers and using it to determine properties like the number of divisors, GCD, or LCM.

• Question

  Find all integer solutions `(k>1) y^k=x^2+x`

  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Algebra -> Equations -> Diophantine Equations
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 1 Points 3

• Question

  Find five natural numbers whose sum is `20`, and whose product is `420`.

  Topics:

  Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample Number Theory -> Prime Numbers -> Prime Factorization
  Sources:
  • problems.ru web site

• Question

  Represent the number `203` as the product of several natural numbers different from `203`, such that the sum of these numbers is also equal to `203`.

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  Topics:

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

• 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`.

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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