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

• 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 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

• JUDKINS'S CATTLE

  Hiram B. Judkins, a cattle-dealer of Texas, had five droves of animals, consisting of oxen, pigs, and sheep, with the same number of animals in each drove. One morning he sold all that he had to eight dealers. Each dealer bought the same number of animals, paying seventeen dollars for each ox, four dollars for each pig, and two dollars for each sheep; and Hiram received in all three hundred and one dollars. What is the greatest number of animals he could have had? And how many would there be of each kind?
  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra -> Word Problems Number Theory -> Division
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 35

• DIGITS AND SQUARES

  It will be seen in the diagram that we have so arranged the nine digits in a square that the number in the second row is twice that in the first row, and the number in the bottom row three times that in the top row. There are three other ways of arranging the digits so as to produce the same result. Can you find them?
  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 77

• THE CAB NUMBERS

  A London policeman one night saw two cabs drive off in opposite directions under suspicious circumstances. This officer was a particularly careful and wide-awake man, and he took out his pocket-book to make an entry of the numbers of the cabs, but discovered that he had lost his pencil. Luckily, however, he found a small piece of chalk, with which he marked the two numbers on the gateway of a wharf close by. When he returned to the same spot on his beat he stood and looked again at the numbers, and noticed this peculiarity, that all the nine digits (no nought) were used and that no figure was repeated, but that if he multiplied the two numbers together they again produced the nine digits, all once, and once only. When one of the clerks arrived at the wharf in the early morning, he observed the chalk marks and carefully rubbed them out. As the policeman could not remember them, certain mathematicians were then consulted as to whether there was any known method for discovering all the pairs of numbers that have the peculiarity that the officer had noticed; but they knew of none. The investigation, however, was interesting, and the following question out of many was proposed: What two numbers, containing together all the nine digits, will, when multiplied together, produce another number (the highest possible) containing also all the nine digits? The nought is not allowed anywhere.
  Topics:

  Arithmetic Combinatorics Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 85

• THE CENTURY PUZZLE

  Can you write `100` in the form of a mixed number, using all the nine digits once, and only once? The late distinguished French mathematician, Edouard Lucas, found seven different ways of doing it, and expressed his doubts as to there being any other ways. As a matter of fact there are just eleven ways and no more. Here is one of them, `91 5742/638`. Nine of the other ways have similarly two figures in the integral part of the number, but the eleventh expression has only one figure there. Can the reader find this last form?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Arithmetic -> Fractions Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 90