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Modular Arithmetic Quiz

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Questions and Answers

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?

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?

Division

Exponentiation (correct)

Square root

Logarithm

What does the notation $a ext{ (mod } n ext{)}$ represent?

The remainder when a is divided by n Signup and view all the answers

If 7:00 is represented as 7 modulo 12, what is the representation of 19:00 modulo 12?

0 Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

What type of domain is the Chinese remainder theorem true over?

It is true over every principal ideal domain. Signup and view all the answers

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.