Number Theory, Modular Arithmetic / Remainder Arithmetic, Divisibility Rules, Divisibility Rule by 11

The divisibility rule for 11 states that a number is divisible by 11 if the alternating sum of its digits (e.g., `a-b+c-d+...`) is divisible by 11. Questions involve applying this rule to test numbers or find unknown digits.

• SLV LVS BLS

  In the following expression, different letters represent different digits, and identical letters represent identical digits:

  SLV = LVS + BLS

  Find the number SLV.

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rule by 11 Algebra -> Equations Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 3

• 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