Question

find all the quadratic residues of 18 .

Short answer

Answer: The quadratic residues of 18 are 0, 1, 4, 7, 9, and 16.

Step-by-step solution

List the integers modulo 18

Modulo 18, the integers range from 0 to 17. So we have the following set of integers: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}

Calculate the squares of the integers modulo 18

We will take the square of each of these integers modulo 18: 0² ≡ 0 (mod 18), 1² ≡ 1 (mod 18), 2² ≡ 4 (mod 18), 3² ≡ 9 (mod 18), 4² ≡ 16 (mod 18), 5² ≡ 7 (mod 18), 6² ≡ 0 (mod 18), 7² ≡ 1 (mod 18), 8² ≡ 4 (mod 18), 9² ≡ 9 (mod 18), 10² ≡ 16 (mod 18), 11² ≡ 7 (mod 18), 12² ≡ 0 (mod 18), 13² ≡ 1 (mod 18), 14² ≡ 4 (mod 18), 15² ≡ 9 (mod 18), 16² ≡ 16 (mod 18), 17² ≡ 7 (mod 18).

List the quadratic residues modulo 18

Comparing these results, we can see that the quadratic residues modulo 18 are: {0, 1, 4, 7, 9, 16} So the quadratic residues of 18 are 0, 1, 4, 7, 9, and 16.

Key concepts

Modular Arithmetic

Modular arithmetic is a fascinating area of mathematics that involves numbers wrapping around upon reaching a certain value, called the modulus. This is like a clock, which resets to 0 after reaching 12. Imagine the modulus as this reset point. When we talk about integers modulo 18, it means considering integers from 0 up to 17, where 18 loops back to 0. This concept is instrumental in finding quadratic residues, because after squaring each integer, we only care about the remainder when divided by 18.

Modular arithmetic helps simplify calculations by focusing only on these remainders. It is widely used in various fields such as cryptography, coding theory, and computer science. By using modular arithmetic, we can uncover patterns and solve complex problems related to divisibility, like finding quadratic residues. For instance, calculating the squares of integers modulo 18 requires squaring each integer, then reducing it mod 18, clipping it into a tidy range from 0 to 17.

Number Theory

Number theory is the study of properties and relationships between numbers, specifically integers. One of the central interests in number theory is understanding how numbers relate to each other under different operations, such as addition, multiplication, and exponentiation.

A key concept in number theory is understanding quadratic residues, which are numbers that can be expressed as the square of some integer within a given modulus. For example, solving the problem of finding quadratic residues modulo 18 involves determining which numbers between 0 and 17 can be written as the square of an integer, again reduced by modulo 18. This is significant because it reveals structure and patterns within the number set modulo 18.

Number theory dives deep into these inherent properties to tackle problems in mathematics and beyond. Through understanding number theory, we derive concepts like congruences, divisors, primes, and residues—all of which present elegant solutions to seemingly challenging numerical problems.

Integers

Integers form a fundamental building block in mathematics. They include all positive numbers, negative numbers, and zero, such as -3, 0, 1, and so on. When working within modular arithmetic or number theory, integers take on a unique role.

In the context of quadratic residues and the problem at hand, integers modulo 18 are those ranging from 0 to 17, providing the basis for exploring their square properties within the modulus. By squaring these integers and reducing the results modulo 18, we categorize them into residues, meaning the remainder when divided by 18, itself a natural property of integers.
Integers are special since they do not possess any fractional or decimal components. Working with them, particularly in advanced mathematics like modular arithmetic, aids in constructing logical proofs and developing theoretical concepts. Thus, integers are more than just numbers; they are fundamental elements in mathematical theory and its applications.