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.

• 5 but not 7

  How many numbers from 1 to 100 (inclusive) are divisible by 5 but not divisible by 7?
   

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  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 6

• Question

  Consider all the numbers from 1 to 4242. Find the difference between the number of odd numbers divisible by 3
  and the number of numbers divisible by 7 in this range.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Arithmetic -> Fractions Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 5

• Long Division

  Calculate the value of the expression and write it as a decimal fraction:

  `1/(1+1/(1+1/(1+1/(1+1/(1))))`

  Topics:

  Arithmetic -> Fractions Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 7

• Sum of Products of Digits

  Let `P(n)` denote the product of the digits of the number `n`. For example, `P(1948) = 1 * 9 * 4 * 8 = 288`.

  A.

  Calculate `P(1) + P(2) + P(3) + ... + P(2016)`

  B.

  Find the maximum value of `{P(n)} /n`, where `2016 <= n <= 5777`.

  (Solution format: "x, y/z" where y/z is an unreduced fraction. For example "10000, 35/100")

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  Topics:

  Arithmetic Algebra -> Word Problems Combinatorics -> Number Tables Number Theory -> Division
  Sources:
  • Gillis Mathematical Olympiad, 2016-2017 Question 2

• Letter Replacement

  Each letter represents a different digit; whenever a specific letter appears, it is necessarily the same digit.

  Find `B-E/2`

  Given: `AB*C=DE`

  And also `F^D=GF`

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  Topics:

  Number Theory -> Prime Numbers Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic

• 2 or 5 but not 3

  How many two-digit numbers are divisible by 2 or 5, and not divisible by 3?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 6

• How many triangles?

  How many triangles are there in the picture?

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Geometry -> Plane Geometry -> Angle Calculation Number Theory -> Division
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 7

• The Number

  Given a positive integer less than 2000.
  If it is not divisible by 43, then it is divisible by 41,
  If it is not divisible by 53, then it is divisible by 43,
  If it is not divisible by 41, then it is divisible by 53.
  Find the number.

  Topics:

  Number Theory -> Prime Numbers Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 3-4 Question 6
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 4

• Children's Clubs

  In a kindergarten, there are three clubs: Judo, Agriculture, and Mathematics. Each child participates in exactly one club, and each club has at least one participant. The total number of children in the kindergarten is 32. On Friday, the kindergarten teacher gathered 6 children to tidy up the classroom. The teacher counted and found that exactly half of the Judo club members, a quarter of the Agriculture club members, and an eighth of the Mathematics club members volunteered for the task. How many students are in each club?

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  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Fractions Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division
  Sources:
  • Gillis Mathematical Olympiad, 2018-2019 Question 1

• The Number with the Most Divisors

  Among the positive integers less than 1000, which number has the most divisors?

  Topics:

  Arithmetic Number Theory -> Prime Numbers -> Prime Factorization Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Number Theory -> Division
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 3