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.

Prime Numbers Chinese Remainder Theorem Modular Arithmetic / Remainder Arithmetic Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Triangular Numbers Division

• SIMPLE MULTIPLICATION

  If we number six cards `1, 2, 4, 5, 7`, and `8`, and arrange them on the table in this order:—

  `1\ \ \ 4\ \ \ 2\ \ \ 8\ \ \ 5\ \ \ 7`

  We can demonstrate that in order to multiply by `3` all that is necessary is to remove the `1` to the other end of the row, and the thing is done. The answer is `428571`. Can you find a number that, when multiplied by `3` and divided by `2`, the answer will be the same as if we removed the first card (which in this case is to be a `3`) From the beginning of the row to the end?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 126

• A CALENDAR PUZZLE

  If the end of the world should come on the first day of a new century, can you say what are the chances that it will happen on a Sunday?
  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 416

• Question

  All the even numbers from `12` to `34` are written on the board without spaces. As a result, the following number was obtained:

  `121416182022242628303234`

  Is this number divisible by `24`?

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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
  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

  How many numbers between `1` and `100` are not divisible by `2` or `5`?

  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 5 and 25 Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Set Theory Number Theory -> Division

• Question

  A two-digit number is written on the board. Avi claims that the units digit of the number is twice the tens digit. Beni claims that the number is divisible by `9`. Gal claims that the number is divisible by `4`. Dani claims that the number is divisible by `27`. It is known that one of them is wrong, and the rest are correct. What number is written on the board?

  Topics:

  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 Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Magic Number 15

  Yossi writes the number `15` on the board. Then, Danny adds a digit to the right and a digit to the left of the number written on the board, such that the new number is still divisible by `15`.

  Find this number. Is there only one possibility?

  Note: The digit added to the left is not zero.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Proof and Example -> Constructing an Example / Counterexample Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 5 and 25 Number Theory -> Division
  Sources:
  • problems.ru web site

• Question

  Is the number  `10^2016+8` divisible by `9`? Justify your answer!

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9
  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

  What is the smallest four-digit number whose first digit is `8`, the number is divisible by `3`, and all its digits are different?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Division