Modular Arithmetic and Chinese Zodiac Cycle
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Questions and Answers

What is the Chinese zodiac animal for the year 2024?

  • Dog
  • Rooster
  • Dragon (correct)
  • Rabbit

If you were born in 1995, which Chinese zodiac animal do you belong to?

  • Rabbit
  • Boar
  • Pig (correct)
  • Ox

What time should you set your alarm if you go to bed at 11:00 PM aiming for 10 hours of sleep?

  • 10:00 AM
  • 8:00 AM
  • 7:00 AM
  • 9:00 AM (correct)

What are the possible remainders when integers are divided by 4?

<p>0, 1, 2, 3 (A)</p> Signup and view all the answers

If you calculate the Chinese zodiac year for 1990, what animal will you find?

<p>Monkey (C)</p> Signup and view all the answers

Which of the following years would correspond to the Year of the Ox?

<p>2009 (B)</p> Signup and view all the answers

What does the expression $y = 9(x - 2) \mod 26$ represent in the context of decoding?

<p>The decoding of numbers back into letters in Z26. (D)</p> Signup and view all the answers

What is the decoded letter for the numerical value 15 using the provided formula?

<p>M (B)</p> Signup and view all the answers

What will be the result of the expression $9(5 - 2) \mod 26$?

<p>1 (A)</p> Signup and view all the answers

What does the letter 'Z' correspond to after using the formula $y = 9(x - 2) \mod 26$?

<p>H (B)</p> Signup and view all the answers

Which step in the decoding process involves calculating $9(-2) \mod 26$?

<p>Decoding the letter 'Z' (C)</p> Signup and view all the answers

What process describes the transformation of plaintext into ciphertext?

<p>Encryption (D)</p> Signup and view all the answers

What is the encoded form of the letter M using the formula $y \equiv 3x + 2 \ (mod \ 26)$?

<p>O (A)</p> Signup and view all the answers

What is the correct modular mathematical operation for encoding A, which has the value of 1?

<p>y = 3(1) + 2 = 5 (mod 26) (B)</p> Signup and view all the answers

Using the encoding formula, what is the result of encoding the letter T, which is represented by the value 20?

<p>J (C)</p> Signup and view all the answers

What value does the letter H become when encoded using the formula $y \equiv 3x + 2 \ (mod \ 26)$?

<p>Z (D)</p> Signup and view all the answers

To decrypt the message, which operation should be performed first according to the decryption steps?

<p>Interchange x and y (A)</p> Signup and view all the answers

After interchanging x and y in the decryption process, what is the next equation that needs to be solved?

<p>x = 3y + 2 (B)</p> Signup and view all the answers

What must be calculated to find y after rearranging to $x - 2 = 3y$?

<p>Subtract 2 from x and divide by 3 (C)</p> Signup and view all the answers

What is the result of adding two odd numbers?

<p>Even (A)</p> Signup and view all the answers

When multiplying an odd number by an even number, what is the result?

<p>Even (B)</p> Signup and view all the answers

In modular arithmetic, what does the notation 1 ⊗ 1 ≡ 1 (mod 2) signify?

<p>Odd number (A)</p> Signup and view all the answers

How does even plus odd compare when considering the result?

<p>Results in Odd (A)</p> Signup and view all the answers

What set describes the possible remainders when a number is divided by a fixed modulus m?

<p>Zm (B)</p> Signup and view all the answers

Which operation is represented by the notation ⊕?

<p>Addition (C)</p> Signup and view all the answers

What is the result when zero is added to any number in the modular system?

<p>The same number (C)</p> Signup and view all the answers

If 1 ⊕ 0 ≡ 1 (mod 2), what does this tell us about the addition of odd and even?

<p>Odd plus even equals odd (D)</p> Signup and view all the answers

Which statement is true concerning the partition of integers based on remainders?

<p>All integers can belong to only one set in the partition. (A)</p> Signup and view all the answers

If the divisor is 3, what are the possible distinct remainders?

<p>0, 1, and 2 (C)</p> Signup and view all the answers

What is the set of integers that gives a remainder of 0 when divided by 3?

<p>{..., -6, -3, 0, 3, 6, ...} (B)</p> Signup and view all the answers

Which of the following would NOT be included in the set for remainder 1 when divided by 3?

<p>12 (D)</p> Signup and view all the answers

Which of the following numbers belongs to the set of integers that produce a remainder of 2 when divided by 3?

<p>23 (D)</p> Signup and view all the answers

What is the result of the operation $8 ⊕ 7$ in $Z_9$?

<p>6 (A)</p> Signup and view all the answers

Which of the following correctly represents the multiplication operation in $Z_2$?

<p>0 ⊗ 1 ≡ 0 (D)</p> Signup and view all the answers

How does the Caesar Cipher function as an encryption technique?

<p>Each letter is replaced by a letter further down the alphabet by a fixed number. (B)</p> Signup and view all the answers

What elements are in the set $Z_3$?

<p>{0, 1, 2} (B)</p> Signup and view all the answers

In the addition table for $Z_2$, what is the result of $1 ⊕ 1$?

<p>0 (B)</p> Signup and view all the answers

Which operation in modulo arithmetic results in a value of $0$ when performed with $3$ in $Z_4$?

<p>3 ⊕ 2 (B)</p> Signup and view all the answers

Flashcards

Chinese Zodiac Cycle

A twelve-year cycle, each year associated with an animal sign.

Zodiac Animal for 2024

2024 divided by 12 leaves a remainder of 8, which corresponds to the Dragon.

Finding your Zodiac Sign

Divide your birth year by 12. The remainder indicates your sign.

Clock Arithmetic

Using modular arithmetic to find times based on a 12-hour clock

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Modular Arithmetic (Clock)

Finding time after sleeping using the remainder when dividing.

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Partitioning Integers

Dividing integers into sets based on their remainders when divided by a specific number.

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Remainders in Modular Arithmetic

The possible results when an integer is divided by another.

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Divisor and Remainder

When a number is divided by another, the leftover portion is the remainder.

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Odd + Odd

The sum of two odd numbers always results in an even number.

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Odd + Even

The sum of an odd and even number always results in an odd number.

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Even + Odd

The sum of an even and odd number always results in an odd number.

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Even + Even

The sum of two even numbers always results in an even number.

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Odd x Odd

The product of two odd numbers always results in an odd number.

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Odd x Even

The product of an odd number and an even number always results in an even number.

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Even x Odd

The product of an even number and an odd number always results in an even number.

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Even x Even

The product of two even numbers always results in an even number.

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Remainder

The leftover amount after dividing one number by another.

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Modulus (m)

The divisor used in modular arithmetic.

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Congruence Modulo m

Two integers are congruent modulo m if they have the same remainder when divided by m.

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What does it mean for two integers to be congruent?

Two numbers are congruent if they have the same remainder when divided by the same number.

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Integer Sets

Sets of integers that share the same remainder

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Finding a Set

Given an integer and a modulo, determine which set it belongs to.

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Why does a set have the same remainder?

All elements within a set have the same remainder when divided by the modulo, which is why they are grouped together.

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Modular Addition

Adding two numbers in a modular system, finding the remainder when the sum is divided by the modulus.

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Modular Multiplication

Multiplying two numbers in a modular system, finding the remainder when the product is divided by the modulus.

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Z_m (Modular System)

A set of numbers {0, 1, 2, ... , m-1} where addition and multiplication are performed modulo m.

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What does 'a ≡ b (mod m)' mean?

The numbers 'a' and 'b' have the same remainder when divided by 'm'.

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Z_2 (Modular System)

The modular system with two numbers {0, 1}, representing even and odd numbers respectively.

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Caesar Cipher

An encryption method where each letter in a message is shifted down the alphabet by a fixed number.

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Encryption

The process of transforming information into a code that is difficult to decipher.

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Plain Text

The original message before encryption.

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Ciphertext

The encrypted message after applying the Caesar cipher.

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Encryption Formula

A mathematical rule used to transform plaintext into ciphertext.

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Modular Arithmetic (mod)

A system where you find the remainder after dividing by a specific number.

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Encoding Process (Caesar Cipher)

Using the encryption formula to transform each letter of plaintext into ciphertext.

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Decryption (Caesar Cipher)

Reversing the encryption process to transform ciphertext back into plaintext.

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Inverse Formula

A mathematical formula used to reverse the encryption process and decode ciphertext.

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Modular Arithmetic

A system of arithmetic where numbers 'wrap around' after reaching a specific value, like a clock. The remainder after division is the key result.

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Inverse in Z26

In Z26, the inverse of a number is another number that, when multiplied, gives a remainder of 1 after dividing by 26. It acts like a reciprocal but in modulo 26.

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Encoding with Inverse

Using the inverse of a number in Z26, we can encode messages by replacing each letter with its corresponding number, then applying the inverse formula.

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Decoding with Inverse Formula

Decoding a message encrypted using an inverse formula in Z26 involves applying the same formula but in reverse, to recover the original letters.

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Modular Arithmetic in Cryptography

Modular arithmetic is used in cryptography to create codes and ciphers that are difficult to break due to the 'wrap-around' nature of the system.

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Study Notes

Modular Arithmetic

  • Modular arithmetic is a system of arithmetic in which numbers "wrap around" upon reaching a certain value, called the modulus.
  • Key concept: Congruence modulo m (denoted as a ≡ b (mod m)).
  • Congruence Definition: Two integers, a and b, are congruent modulo m if their difference (a - b) is divisible by m. Alternatively, a and b have the same remainder when divided by m.

Chinese Zodiac Cycle

  • The Chinese zodiac cycle follows a twelve-year cycle.
  • Each year is associated with a specific animal sign.
  • Cycle order: Monkey, Rooster, Dog, Pig, Rat, Ox, Tiger, Rabbit, Dragon, Snake, Horse, Sheep.
  • To find the zodiac animal for a given year, divide the year by 12; the remainder gives the position in the cycle. For example, 2024 divided by 12 has a remainder of 8, so 2024 is the year of the Dragon.

Clock Arithmetic

  • Clock arithmetic uses the idea of remainders when dividing by 12.
  • Example Calculation: If you go to bed at 11:00 PM and need 10 hours of sleep, your alarm should be set for 9:00 AM.

Partition of Integers

  • Integers can be partitioned into disjoint sets based on their remainders when divided by a specific divisor.
  • Example: If the divisor is 3, the remainders are 0, 1, and 2. Each integer will fall into one of these sets.

Set of Remainder 0, 1 & 2 (Divisor = 3)

  • Remainder 0: {..., -6, -3, 0, 3, 6, 9, 12,...}
  • Remainder 1: {..., -5, -2, 1, 4, 7, 10, 13,...}
  • Remainder 2: {..., -4, -1, 2, 5, 8, 11, 14,...}

Congruence Modulo m

  • Fix a positive integer m (called the modulus). For integers a and b, a ≡ b (mod m) if a - b is divisible by m, or equivalently, a and b have the same remainder when divided by m.

Example: Congruence Modulo

  • 29 ≡ 8 (mod 3) because (29 - 8 = 21) is divisible by 3.

Exercise: Congruence Modulo Examples

  • Provide sample problems involving modular arithmetic congruences.

Odd vs Even

  • Odd integers are congruent to 1 (mod 2).
  • Even integers are congruent to 0 (mod 2).

Odd-Even Arithmetic

  • Sum Rules: Odd + Odd = Even, Odd + Even = Odd, Even + Even = Even
  • Product Rules: Odd × Odd = Odd, Odd × Even = Even, Even × odd = Even, Even × Even = Even

Modular Number System

  • For a fixed modulus m, the set Zm = {0, 1, 2, ..., m – 1}.
  • Zm represents the possible remainders when a number is divided by m.
  • Defined Operations on Zm: Addition modulo m, Multiplication modulo m.

Caesar Cipher

  • Encryption method where each letter in a message is shifted a fixed number of positions down the alphabet.
  • Example: Plaintext "HELLO" becomes "IFMMP" with a shift of 1 (replace A with B, B with C etc.).

Generalized Caesar Cipher

  • Used in situations where the alphabet is represented numerically (ex. A=1, B=2, Z=25).
  • Modular arithmetic is applied to encode and decode. The letters are mapped to numbers {0,1,...,25).

Encryption

  • In cryptography, encryption is the process of converting plaintext to ciphertext for security.
  • Example: Secret message MATH, encoded by y=3x+2 (mod 26) results in OEJZ.

Decryption

  • The process used to decode the ciphertext into its original plaintext.
  • You find the inverse formula for solving the encrypted calculation.

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Description

Explore the fascinating concepts of modular arithmetic and the Chinese zodiac cycle. This quiz covers congruence, the significance of the modulus, and how to determine zodiac signs based on the year. Test your understanding of these mathematical and cultural concepts!

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