Podcast
Questions and Answers
What is the congruence relation denoted by modulo n?
What is the congruence relation denoted by modulo n?
- $a + b ext{ (mod } n ext{)}$
- $a ext{ mod } n = b$
- $a eq b ext{ (mod } n ext{)}$ (correct)
- $a imes b ext{ (mod } n ext{)}$
What does the statement $15
eq 3 ext{ (mod } 12 ext{)}$ mean?
What does the statement $15 eq 3 ext{ (mod } 12 ext{)}$ mean?
- 15 is not congruent to 3 modulo 12 (correct)
- 15 is congruent to 3 modulo 12
- 15 is a multiple of 3 modulo 12
- 15 is a divisor of 3 modulo 12
Which operation is congruence modulo n compatible with?
Which operation is congruence modulo n compatible with?
- Division
- Exponentiation (correct)
- Square root
- Logarithm
What does the notation $a ext{ (mod } n ext{)}$ represent?
What does the notation $a ext{ (mod } n ext{)}$ represent?
If 7:00 is represented as 7 modulo 12, what is the representation of 19:00 modulo 12?
If 7:00 is represented as 7 modulo 12, what is the representation of 19:00 modulo 12?
What does the Chinese remainder theorem state?
What does the Chinese remainder theorem state?
What is the earliest known statement of the Chinese remainder theorem?
What is the earliest known statement of the Chinese remainder theorem?
What does the Chinese remainder theorem allow for computing with large integers?
What does the Chinese remainder theorem allow for computing with large integers?
In the context of the Chinese remainder theorem, what is the significance of the divisors being pairwise coprime?
In the context of the Chinese remainder theorem, what is the significance of the divisors being pairwise coprime?
What type of domain is the Chinese remainder theorem true over?
What type of domain is the Chinese remainder theorem true over?
Study Notes
Congruence Modulo n
- Congruence modulo n is a congruence relation denoted by ≡ (mod n)
- The statement 15 ≡ 3 (mod 12) means that 15 and 3 have the same remainder when divided by 12
- Congruence modulo n is compatible with the addition and multiplication operations
Notation and Representation
- The notation a (mod n) represents the remainder of a when divided by n
- In modular arithmetic, 7:00 is represented as 7 modulo 12, and 19:00 is represented as 7 modulo 12
Chinese Remainder Theorem
- The Chinese remainder theorem states that if we have a system of congruences: x ≡ a1 (mod n1) x ≡ a2 (mod n2) ... x ≡ ak (mod nk) where ni are pairwise coprime, then there exists a unique solution modulo N = n1n2...nk
- The earliest known statement of the Chinese remainder theorem dates back to the 4th century AD, in the book "Sun Zi Suan Jing" by Sun Zi
- The Chinese remainder theorem allows for efficient computing with large integers by breaking them down into smaller components
- The significance of the divisors being pairwise coprime is that it ensures the existence of a unique solution
- The Chinese remainder theorem is true over the domain of integers
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Description
Test your knowledge of modular arithmetic with this quiz! Explore the concept of numbers "wrapping around" a modulus, as well as the applications of modular arithmetic in various systems.
