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.

• Pumbaa and the Candies

  Pumbaa has 11 chocolate candies and 13 toffee candies. Each time he can eat either two candies of different types,
  or three candies of the same type. What is the largest number of candies that Pumbaa can eat according to these rules?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Combinatorics -> Invariants Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 6

• Question

  A two-digit number is written on the board.
  Avi said: "The units digit of the number is 3"
  Beni said: "It's a square number"
  Gili said: "This number is a multiple of 12"
  Then the teacher said: "There are two correct statements and one wrong one here."
  What number was written on the board?

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Logic -> Truth-tellers and Liars Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 1

• Question

  Vered subtracted a number composed of the same digits written in reverse order from a three-digit number.
  As a result, she obtained a two-digit number. Find the two-digit number she obtained.

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra Logic -> Reasoning / Logic
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 2

• 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

• Question

  A two-digit number is written on the board.
  Avi said: "The digit 5 appears in this number."
  Beni said: "This is a square number."
  Gili said: "This number is greater than 50."
  Dani said: "The number is divisible by 7."
  Then the teacher said: "There are three correct statements here and one incorrect statement.".
  What number was written on the board?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 1

• Consecutive Numbers

  a. Avi wants to find 10 consecutive numbers whose sum is divisible by 90. Will he succeed?
  b. Benny wants to find 11 consecutive numbers such that their sum is divisible by 90. Will he succeed?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 4

• 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

• The Units Digit

  Miriam has eight cards with consecutive three-digit numbers. The units digit of the smallest number is 1,
  the units digit of the largest number is 8. Miriam arranged the cards in a row such that the first number is divisible by 2,
  the second number is divisible by 3, the third number is divisible by 4, and so on until the eighth number which is divisible by 9.
  What is the units digit of the number divisible by 7?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9 Question 6

• Numbers on a Board

  At the beginning of the day, four integers are written on the board (`a_0,b_0,c_0,d_0`). Every minute, Danny replaces the four numbers on the board with a new set of four numbers according to the following rule: If the numbers written on the board are (a,b,c,d), Danny first generates the numbers
  `a'=a+4b+16c+64d`
  `b'=b+4c+16d+64a`
  `c'=c+4d+16a+64b`
  `d'=d+4a+16b+64c`
  Then he erases the numbers (a,b,c,d) and writes in their place the numbers (a',d',c',b'). For which initial sets (`a_0,b_0,c_0,d_0`) will Danny eventually write a set of four numbers that are all divisible by `5780^5780`

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 4

• Finite Division

  Find all integers x, y, z, w that satisfy `x^2+y^2=3z^2+3w^2`.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Proof and Example Algebra -> Equations -> Diophantine Equations