Number Theory, Modular Arithmetic / Remainder Arithmetic, Divisibility Rules

Divisibility rules are shortcuts to determine if an integer is exactly divisible by another smaller integer without performing the full division. Questions involve applying these rules for various divisors (e.g., 2, 3, 4, 5, 9, 10, 11) to test numbers or find missing digits.

• 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

• Question

  Take any three-digit number and arbitrarily rearrange its digits. Prove that the difference between the original number and the new number is divisible by `3`. Is this also true for four-digit numbers?

  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 natural number which, when divided by the sum of its digits with a remainder, yields `2017` as both the quotient and the remainder?

  Topics:

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

• Question

  Prove that the sum of the digits of a perfect square cannot be equal to `2019 `.

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  Topics:

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

• Question

  Given two numbers such that one is obtained from the other by changing the order of its digits. Prove that their difference is divisible by `9`.

  Topics:

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

• Question

  Given a natural number `n` that is three times greater than the sum of its digits. Prove that `n` is divisible by `27`.

  Topics:

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