Arithmetic, Division with Remainder

This topic focuses on the division of integers where the result includes a quotient and a remainder (which is less than the divisor). Questions involve performing such divisions and solving problems where the remainder is significant, such as in modular arithmetic or pattern recognition.

• Question

  Avi lives in apartment number `45`. On which floor is the apartment located if there are `2` apartments on the first floor of his building, and `4` apartments on every other floor? (Note: The building does not have a floor `0`)

  Topics:

  Algebra -> Word Problems Arithmetic -> Division with Remainder
  Sources:
  • problems.ru web site

• Question

  Find all natural numbers with the following property: when divided by 7, their remainder is equal to their quotient.

  Topics:

  Number Theory Algebra -> Inequalities Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Arithmetic -> Division with Remainder

• Question

  Prove that among five integers, it is possible to choose two whose difference is divisible by `4`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Combinatorics -> Pigeonhole Principle Number Theory -> Division -> Parity (Even/Odd) Arithmetic -> Division with Remainder

• 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

  Avi invited guests. They know he lives in apartment `333`, in entrance number `10`, but they don't know which floor. There are `9` floors in the building, and it is known that there are an equal number of apartments on each floor. The guests want to use the elevator. Can they calculate in advance which floor they need to press in the elevator?

  Topics:

  Algebra -> Inequalities Algebra -> Word Problems Arithmetic -> Division with Remainder

• 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

  Is there a solution in natural numbers to the equation `x^2 + 12 = y^3` such that

  a. x is even (easier)

  b. x is odd

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  Topics:

  Number Theory -> Prime Numbers Number Theory -> Modular Arithmetic / Remainder Arithmetic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction Arithmetic -> Division with Remainder Algebra -> Equations -> Diophantine Equations
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 6

• Log of Wood

  You have a very long log of wood. Can you measure exactly one meter from it, if you have for this purpose:
  а. A stick with a length of one and a half meters and another stick with a length of 40 centimeters,
  б. A stick with a length of one and a half meters and another stick with a length of 30 centimeters,

  Assuming you have no other measuring tools? Explain!

  Topics:

  Combinatorics -> Invariants Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Division with Remainder
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 7
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 2

• Consecutive Numbers and Deletion

  Yossi wrote 10 consecutive natural numbers on the board. Danny erased one of the numbers. The sum of the remaining numbers on the board is 2020.
  Which number was erased?

  Topics:

  Algebra -> Word Problems Algebra -> Sequences Arithmetic -> Division with Remainder
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 5

• THE MILLIONAIRE'S PERPLEXITY

  Mr. Morgan G. Bloomgarten, the millionaire, known in the States as the Clam King, had, for his sins, more money than he knew what to do with. It bored him. So he determined to persecute some of his poor but happy friends with it. They had never done him any harm, but he resolved to inoculate them with the "source of all evil." He therefore proposed to distribute a million dollars among them and watch them go rapidly to the bad. But he was a man of strange fancies and superstitions, and it was an inviolable rule with him never to make a gift that was not either one dollar or some power of seven—such as `7, 49, 343, 2,401`, which numbers of dollars are produced by simply multiplying sevens together. Another rule of his was that he would never give more than six persons exactly the same sum. Now, how was he to distribute the `1,000,000` dollars? You may distribute the money among as many people as you like, under the conditions given.
  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Arithmetic -> Division with Remainder
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 16