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

  Given a three-digit prime number with all its digits distinct. It is known that its last digit is equal to the sum of the other two digits. Find all the possibilities for the last digit of this number.

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  Topics:

  Number Theory -> Prime Numbers 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 Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  Find all pairs of prime numbers whose difference is `17`.

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  Topics:

  Number Theory -> Prime Numbers Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Question

  Given the polynomial `P(n)=n^2+n+41`. Is it true that this polynomial yields prime numbers for all natural numbers `n`?

  Topics:

  Number Theory -> Prime Numbers Algebra Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • problems.ru web site

• Question

  Is the following number prime? 

  `4^9 + 6^10 + 3^20`

  Topics:

  Number Theory -> Prime Numbers Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities

• Question

  Prove that if `n!+1` is divisible by `n+1`, then `n+1` is prime.

  Topics:

  Number Theory -> Prime Numbers Proof and Example -> Proof by Contradiction

• Question

  Does there exist an infinite arithmetic progression consisting only of prime numbers?

  Note: We do not consider "trivial" arithmetic progressions, which are constant.

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  Topics:

  Number Theory -> Prime Numbers Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Proof and Example -> Proof by Contradiction

• 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

• Question

  Given `50` distinct natural numbers between `1` and `100`. It is known that no two of these numbers sum to `100`. Is it necessarily true that one of these numbers must be a perfect square?

  Topics:

  Number Theory -> Prime Numbers Arithmetic Combinatorics -> Pigeonhole Principle Combinatorics -> Matchings Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Set Theory Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction

• Question

  Is there a solution in natural numbers to the equation `x^2 + 12 = y^3` such that

  a. x is even (easier)

  b. x is odd

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  Topics:

  Number Theory -> Prime Numbers Number Theory -> Modular Arithmetic / Remainder Arithmetic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction Arithmetic -> Division with Remainder Algebra -> Equations -> Diophantine Equations
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 6

• Letter Replacement

  Each letter represents a different digit; whenever a specific letter appears, it is necessarily the same digit.

  Find `B-E/2`

  Given: `AB*C=DE`

  And also `F^D=GF`

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  Topics:

  Number Theory -> Prime Numbers Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic