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What is the Chinese zodiac animal for the year 2024?
What is the Chinese zodiac animal for the year 2024?
If you were born in 1995, which Chinese zodiac animal do you belong to?
If you were born in 1995, which Chinese zodiac animal do you belong to?
What time should you set your alarm if you go to bed at 11:00 PM aiming for 10 hours of sleep?
What time should you set your alarm if you go to bed at 11:00 PM aiming for 10 hours of sleep?
What are the possible remainders when integers are divided by 4?
What are the possible remainders when integers are divided by 4?
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If you calculate the Chinese zodiac year for 1990, what animal will you find?
If you calculate the Chinese zodiac year for 1990, what animal will you find?
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Which of the following years would correspond to the Year of the Ox?
Which of the following years would correspond to the Year of the Ox?
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What does the expression $y = 9(x - 2) \mod 26$ represent in the context of decoding?
What does the expression $y = 9(x - 2) \mod 26$ represent in the context of decoding?
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What is the decoded letter for the numerical value 15 using the provided formula?
What is the decoded letter for the numerical value 15 using the provided formula?
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What will be the result of the expression $9(5 - 2) \mod 26$?
What will be the result of the expression $9(5 - 2) \mod 26$?
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What does the letter 'Z' correspond to after using the formula $y = 9(x - 2) \mod 26$?
What does the letter 'Z' correspond to after using the formula $y = 9(x - 2) \mod 26$?
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Which step in the decoding process involves calculating $9(-2) \mod 26$?
Which step in the decoding process involves calculating $9(-2) \mod 26$?
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What process describes the transformation of plaintext into ciphertext?
What process describes the transformation of plaintext into ciphertext?
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What is the encoded form of the letter M using the formula $y \equiv 3x + 2 \ (mod \ 26)$?
What is the encoded form of the letter M using the formula $y \equiv 3x + 2 \ (mod \ 26)$?
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What is the correct modular mathematical operation for encoding A, which has the value of 1?
What is the correct modular mathematical operation for encoding A, which has the value of 1?
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Using the encoding formula, what is the result of encoding the letter T, which is represented by the value 20?
Using the encoding formula, what is the result of encoding the letter T, which is represented by the value 20?
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What value does the letter H become when encoded using the formula $y \equiv 3x + 2 \ (mod \ 26)$?
What value does the letter H become when encoded using the formula $y \equiv 3x + 2 \ (mod \ 26)$?
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To decrypt the message, which operation should be performed first according to the decryption steps?
To decrypt the message, which operation should be performed first according to the decryption steps?
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After interchanging x and y in the decryption process, what is the next equation that needs to be solved?
After interchanging x and y in the decryption process, what is the next equation that needs to be solved?
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What must be calculated to find y after rearranging to $x - 2 = 3y$?
What must be calculated to find y after rearranging to $x - 2 = 3y$?
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What is the result of adding two odd numbers?
What is the result of adding two odd numbers?
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When multiplying an odd number by an even number, what is the result?
When multiplying an odd number by an even number, what is the result?
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In modular arithmetic, what does the notation 1 ⊗ 1 ≡ 1 (mod 2) signify?
In modular arithmetic, what does the notation 1 ⊗ 1 ≡ 1 (mod 2) signify?
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How does even plus odd compare when considering the result?
How does even plus odd compare when considering the result?
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What set describes the possible remainders when a number is divided by a fixed modulus m?
What set describes the possible remainders when a number is divided by a fixed modulus m?
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Which operation is represented by the notation ⊕?
Which operation is represented by the notation ⊕?
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What is the result when zero is added to any number in the modular system?
What is the result when zero is added to any number in the modular system?
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If 1 ⊕ 0 ≡ 1 (mod 2), what does this tell us about the addition of odd and even?
If 1 ⊕ 0 ≡ 1 (mod 2), what does this tell us about the addition of odd and even?
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Which statement is true concerning the partition of integers based on remainders?
Which statement is true concerning the partition of integers based on remainders?
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If the divisor is 3, what are the possible distinct remainders?
If the divisor is 3, what are the possible distinct remainders?
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What is the set of integers that gives a remainder of 0 when divided by 3?
What is the set of integers that gives a remainder of 0 when divided by 3?
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Which of the following would NOT be included in the set for remainder 1 when divided by 3?
Which of the following would NOT be included in the set for remainder 1 when divided by 3?
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Which of the following numbers belongs to the set of integers that produce a remainder of 2 when divided by 3?
Which of the following numbers belongs to the set of integers that produce a remainder of 2 when divided by 3?
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What is the result of the operation $8 ⊕ 7$ in $Z_9$?
What is the result of the operation $8 ⊕ 7$ in $Z_9$?
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Which of the following correctly represents the multiplication operation in $Z_2$?
Which of the following correctly represents the multiplication operation in $Z_2$?
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How does the Caesar Cipher function as an encryption technique?
How does the Caesar Cipher function as an encryption technique?
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What elements are in the set $Z_3$?
What elements are in the set $Z_3$?
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In the addition table for $Z_2$, what is the result of $1 ⊕ 1$?
In the addition table for $Z_2$, what is the result of $1 ⊕ 1$?
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Which operation in modulo arithmetic results in a value of $0$ when performed with $3$ in $Z_4$?
Which operation in modulo arithmetic results in a value of $0$ when performed with $3$ in $Z_4$?
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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!
