Answer the following question: Information: - The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book, Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own. - An integer (from the Latin "integer" meaning "whole") is a number that can be written without a fractional component. For example, 21, 4, 0, and2048 are integers, while 9.75, , and are not. - In mathematics, especially in set theory, a set "A" is a subset of a set "B", or equivalently "B" is a superset of "A", if "A" is "contained" inside "B", that is, all elements of "A" are also elements of "B". "A" and "B" may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. - In Algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in formula_1 (the set of integers). The set of all algebraic integers, , is closed under addition and multiplication and therefore is a commutative subring of the complex numbers. The ring is the integral closure of regular integers formula_1 in complex numbers. - Johann Carl Friedrich Gauss (30 April 1777 Braunschweig23 February 1855 Gottingen) was a German mathematician who contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, mechanics, electrostatics, astronomy, matrix theory, and optics. - In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain valuethe modulus (plural moduli). The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book "Disquisitiones Arithmeticae", published in 1801. - A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 3, 1 1 3, etc. are all valid factorizations of 3. - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. - In mathematics, a rational number is any number that can be expressed as the quotient or fraction "p"/"q" of two integers, a numerator "p" and a non-zero denominator "q". Since "q" may be equal to1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", is usually denoted by a boldface Q (or blackboard bold formula_1, Unicode ); it was thus denoted in 1895 by Giuseppe Peano after "quoziente", Italian for "quotient". - A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit1, so the composite numbers are exactly the numbers that are not prime and not a unit. - Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers). - Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. This was a recognizable category of mathematical activity from the 19th century onwards, at variance with the trend towards meeting the needs of navigation, astronomy, physics, economics, engineering, and so on. - In number theory , a Carmichael number is a composite number n which satisfies the modular arithmetic congruence relation : b ^ ( n - 1 ) \ equiv 1 \ pmod ( n ) for all integers 1 < b < n which are relatively prime to n. They are named for Robert Carmichael . The Carmichael numbers are the subset K1 of the Knodel numbers . - Arithmetic or arithmetics (from the Greek "arithmos", "number") is the oldest and most elementary branch of mathematics. It consists of the study of numbers, especially the properties of the traditional operations between themaddition, subtraction, multiplication and division. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms "arithmetic" and "higher arithmetic" were used until the beginning of the 20th century as synonyms for "number theory" and are sometimes still used to refer to a wider part of number theory. After reading the paragraphs above, choose the best answer for the entity that related to 'carmichael number' with the relationship of 'subclass of'. Choices: - activity - algebraic number - analysis - astronomy - blackboard - branch - change - complex number - definition - division - economics - engineering - geometry - geophysics - german - greek - integer - latin - learning - logic - mathematics - meeting - natural number - number - object - optics - own - part - plural - prime number - quantity - range - rational number - ring - role - set - set theory - space - statistics - structure - subset - system - theory
Answer:
natural number