Number Theory

Number Theory is a branch of mathematics concerned with the properties of integers. Topics include prime numbers, divisibility, congruences (modular arithmetic), Diophantine equations, and functions of integers. Questions often require analytical and creative thinking about numbers.

Prime Numbers Chinese Remainder Theorem Modular Arithmetic / Remainder Arithmetic Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Triangular Numbers Division

• Cherries and Blueberries

  `175` kg of cherries cost more than `125` kg of blueberries, but less than `126` kg of blueberries. In addition, it is known that a kilogram of cherries costs a whole number of shekels, and a kilogram of blueberries also costs a whole number of shekels.

  Prove that `80` shekels is not enough to buy one kilogram of blueberries and three kilograms of cherries.

  S. Fomin

  Topics:

  Number Theory Arithmetic Algebra -> Inequalities Algebra -> Word Problems
  Sources:
  • Tournament of Towns, 1983-1984, Fall, Practice Version, Grades 9-10 Question 1 Points 3

• Product of Digits

  For a natural number `x`, let `P(x)` denote the product of the digits of `x`, and let `S(x)` denote the sum of the digits of `x`. 

  How many solutions are there to the equation:

  `P(P(x))+P(S(x))+S(P(x))+S(S(x))=1984`

  Topics:

  Number Theory Algebra -> Equations
  Sources:
  • Tournament of Towns, 1983-1984, Fall, Practice Version, Grades 9-10 Question 4 Points 8

• Question

  The numbers `1`, `2`, `3`, ..., `9` are divided into `3` sets. Prove that there is a set where the product of the numbers is greater than or equal to `72`.

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  Topics:

  Number Theory Combinatorics -> Pigeonhole Principle Algebra -> Inequalities Proof and Example -> Proof by Contradiction

• Question

  The numbers a, b, c are different from 0, and the numbers ab, ac, bc are rational.

  a. Show that `a^2+b^2+c^2` is a rational number.

  b. Show that if `a^3+b^3+c^3` is a rational number, then `a+b+c` is a rational number.

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  Topics:

  Number Theory Algebra
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 1

• Question

  Given distinct rational numbers a, b, c, prove that `sqrt{1/(a-b)^2+1/(b-c)^2 +1 /(c-a)^2}`

  is rational.

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  Topics:

  Number Theory Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 3

• Digit Sum - 9

  How many numbers in the range from 1 to 500 have a digit sum of 9?

  Topics:

  Number Theory Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 3-4 Question 2

• Question

  Find the smallest two-digit number `☺` that satisfies:

  `☺times☺ - ☺ = 600`

  Topics:

  Number Theory Arithmetic Algebra -> Equations
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 5-6 Question 5

• Ten-Digit Number

  Yael writes ten-digit numbers in whose decimal representation each of the digits `0, 1, 2, 3, 4, 5, 6, 7, 8, 9` appears exactly once.
  In the numbers that Yael writes, the difference between any two adjacent digits is at least 2. What is the smallest number Yael can write?
   

  Topics:

  Number Theory Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 3-4 Question 3
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 1

• Mathematical Conference

  202 participants from three countries attended a mathematical conference: Israel, Greece, and Japan.
  On the first day, every pair of participants from the same country shook hands. On the second day, every pair of participants
  where one was Israeli and the other was not Israeli shook hands. On the third day, every pair of participants where one
  was Israeli and the other was Greek shook hands. In total, 20200 handshakes occurred. How many
  Israeli participants were at the conference?

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  Topics:

  Number Theory Combinatorics Algebra -> Word Problems Algebra -> Equations -> Diophantine Equations
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 2

• The Number Circle

  The numbers 1 through 6 are written in a circle in order, as shown in the figure.
  In each step, Lior chooses a number a in the circle whose neighbors are b and c, and replaces
  it with the number `(bc)/a`.
  Can Lior reach a state where the product of the numbers in the circle is greater than `10^100`
  (a) in 100 steps.
  (b) in 110 steps

  

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  Topics:

  Number Theory Algebra -> Inequalities Algebra -> Sequences
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 6