Number Theory, Division

Division is one of the four basic arithmetic operations, representing the splitting of a quantity into equal parts or finding how many times one number is contained within another. Questions involve performing division, understanding concepts like dividend, divisor, quotient, and remainder, and solving related word problems.

• Clock Tower

  A bell in a clock tower rings `3` times in `12` seconds. How long does it take to ring `6` times?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Number Theory -> Division
  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

• A mistake in the exercise

  Prove that there is an error in the following multiplication problem:

  \(\begin{array}& & & * & * & * & 2 & 7 \\ \times & & & & & * & * \\ \hline & * & * & * & * & * & 6 \\ + & * & * & * & * & * & \\ \hline & * & * & * & * & 4 & 6 \end{array}\)

  Topics:

  Arithmetic Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Number Theory -> Division Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 4

• Birds and Seeds

  Nine identical birds eat less than `1001` seeds for lunch, and ten such birds eat more than `1100` seeds for lunch. How many seeds does one bird eat for lunch?

  Topics:

  Algebra -> Inequalities Algebra -> Word Problems Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division

• 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

  In the magical land, there are only coins of `5`, `6`, and `15` liras. Shlomi currently only has coins of `6` and `15` liras. Shlomi wants to buy a book that costs `38` liras. Will he be able to pay for the book without change?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division

• Question

  In Ella's building, there are three entrances and `13` floors (from `0` to `12`, and the apartments are on every floor except the ground floor). In each entrance, on each floor, there are five apartments. Ella lives in apartment number `73`. In which entrance and on which floor does she live?

  Topics:

  Algebra -> Word Problems Number Theory -> Division Arithmetic -> Division with Remainder

• Question

  A cannibal captured `6` people.

  a. In how many different ways can he choose one person for breakfast, one person for lunch, and one person for dinner?

  b. In how many different ways can he choose three people to release them?

  Topics:

  Combinatorics -> Binomial Coefficients and Pascal's Triangle Combinatorics -> Product Rule / Rule of Product Number Theory -> Division

• Question

  In the magical land, there are only two types of coins: `16` LC (Magical Pounds) and `27` LC. Is it possible to buy a notebook that costs one Magical Pound and receive exact change?

  Topics:

  Proof and Example -> Constructing an Example / Counterexample Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) -> Euclidean Algorithm Number Theory -> Division Algebra -> Equations -> Diophantine Equations

• Reconstruct the Exercise

  Reconstruct the digits in the following exercise:

  

  Topics:

  Arithmetic Number Theory -> Division Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic