10 Questions
What is the congruence relation denoted by modulo n?
$a eq b ext{ (mod } n ext{)}$
What does the statement $15 eq 3 ext{ (mod } 12 ext{)}$ mean?
15 is not congruent to 3 modulo 12
Which operation is congruence modulo n compatible with?
Exponentiation
What does the notation $a ext{ (mod } n ext{)}$ represent?
The remainder when a is divided by n
If 7:00 is represented as 7 modulo 12, what is the representation of 19:00 modulo 12?
0
What does the Chinese remainder theorem state?
It states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.
What is the earliest known statement of the Chinese remainder theorem?
The earliest known statement of the theorem is by the Chinese mathematician Sunzi in the Sunzi Suanjing in the 3rd century CE.
What does the Chinese remainder theorem allow for computing with large integers?
It allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers.
In the context of the Chinese remainder theorem, what is the significance of the divisors being pairwise coprime?
It ensures that the remainder of the division of n by the product of these integers is uniquely determined.
What type of domain is the Chinese remainder theorem true over?
It is true over every principal ideal domain.
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
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.
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