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.

Divisibility Rules by 2, 4, and 8 Divisibility Rules by 3 and 9 Divisibility Rule by 11 Divisibility Rules by 5 and 25

• Question

  Find the largest natural number in which all digits are distinct, and if you look at every 3 consecutive digits, you get a number divisible by 13.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rule by 11 Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Beno Arbel Olympiad, 2017, Grade 8 Question 7

• THE MYSTIC ELEVEN

  Can you find the largest possible number containing any nine of the ten digits (calling nought a digit) that can be divided by `11` without a remainder? Can you also find the smallest possible number produced in the same way that is divisible by `11`? Here is an example, where the digit `5` has been omitted: `896743012`. This number contains nine of the digits and is divisible by `11`, but it is neither the largest nor the smallest number that will work.

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rule by 11
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 93

• Question

  a. Prove that among `11` natural numbers, it is always possible to choose two such that their units digit is the same.

  b. Prove that among `11` natural numbers, it is always possible to choose two such that their difference is divisible by `10`.

  Topics:

  Combinatorics -> Pigeonhole Principle Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 5 and 25