Number Theory, Modular Arithmetic / Remainder Arithmetic

Modular Arithmetic (or Remainder Arithmetic) is a system where numbers 'wrap around' after reaching a certain value, the modulus. It deals with congruences and remainders. Questions involve solving equations in modular systems, finding powers modulo `n`, and applications in patterns or cryptography.

• 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`

  Show hint

  Log in here to receive the hint
  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`.

  Show hint

  Log in here to receive the hint
  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

• SIMPLE MULTIPLICATION

  If we number six cards `1, 2, 4, 5, 7`, and `8`, and arrange them on the table in this order:—

  `1\ \ \ 4\ \ \ 2\ \ \ 8\ \ \ 5\ \ \ 7`

  We can demonstrate that in order to multiply by `3` all that is necessary is to remove the `1` to the other end of the row, and the thing is done. The answer is `428571`. Can you find a number that, when multiplied by `3` and divided by `2`, the answer will be the same as if we removed the first card (which in this case is to be a `3`) From the beginning of the row to the end?

  Topics:

  Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 126

• A CALENDAR PUZZLE

  If the end of the world should come on the first day of a new century, can you say what are the chances that it will happen on a Sunday?
  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 416