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'''Integers''' are the [[natural numbers]] and their negatives.<ref>Negative numbers ha These are some of the integers: ...

...two [[integer]]s is the greatest (largest) number that divides both of the integers evenly. [[Euclid]] came up with the idea of GCDs. The GCD of any two positive integers can be defined as a [[recursive]] [[function (mathematics)|function]]: ...

...satisfies the [[congruence]] <math>b^{n-1}\equiv 1\pmod{n}</math> for all integers <math>b</math> which are [[coprime|relatively prime]] to <math>n</math>. Be ...ber]]s <math>p</math> satisfy <math>b^{p-1}\equiv 1\pmod{p}</math> for all integers <math>b</math> which are relatively prime to <math>p</math>. This has been ...

...n as <math>(\mathbb{Z}, \cdot)</math>. The name of the magma would be "The integers under multiplication". ...

...s, we can say that there are "more" real numbers than integers because the integers are ''countable'' and the real numbers are ''uncountable''. ...ems are inside the real numbers. For example, the [[rational number]]s and integers are all in the real numbers. There are also more complicated number systems ...

...lso called '''clock arithmetic''', is a way of doing [[arithmetic]] with [[integers]]. Much like hours on a [[clock]], which repeat every twelve hours, once th Modular arithmetic can be used to show the idea of '''congruence'''. Two integers, ''a'' and ''b'', are '''congruent modulo n''' if they have the same [[rema ...

...'totient''' of a [[positive number|positive]] [[integer]] is the number of integers smaller than ''n'' which are [[coprime]] to ''n'' (they share no [[factor]] ...because it gives the size of the multiplicative [[group (math)|group]] of integers [[modular arithmetic|modulo]] ''n''. More precisely, <math>\phi(n)</math> i ...

...lly denoted by '''LCM(''a'', ''b'')'''. Likewise, the LCM of more than two integers is the smallest positive integer that is divisible by each of them. ...

...exists a '''nonempty''' subsequence (not necessarily consecutive) of these integers, whose sum is equal to <math> 0 </math>. (Hint: Consider <math> b_i=a_i-i < ...isets''' <math> A </math> and <math> B </math>, both with <math> n </math> integers from <math> 1 </math> to <math> n </math>. Show that there exist two '''non ...

===Integers, addition, zero=== ...of ''infinite order'') because it has an infinite number of elements, the integers. ...

...n the [[natural numbers]], subtraction does not have closure, but in the [[integers]] subtraction does have closure. Subtraction of two numbers can produce a n ...times make closure of a mathematical object by including new elements. The integers are a closure of the natural numbers by including negative numbers. The [[r ...

...exists a '''nonempty''' subsequence (not necessarily consecutive) of these integers, whose sum is equal to <math> 0 </math>. (Hint: Consider <math> b_i=a_i-i < ...isets''' <math> A </math> and <math> B </math>, both with <math> n </math> integers from <math> 1 </math> to <math> n </math>. Show that there exist two '''non ...

The number '''0''' is the "additive identity" for integers, real numbers, and complex numbers. For the real numbers, for all <math>a\i Similarly, The number '''1''' is the "multiplicative identity" for integers, real numbers, and complex numbers. For the real numbers, for all <math>a\i ...

...example, for an equation over the rationals, one can find solutions in the integers. Then, the equation is a [[diophantine equation]]. One may also look for s ...

...agorean triple''' is a set of three [[Positive number|positive]] [[integer|integers]] which satisfy the [[equation]] (make the equation work): ...

Ramanujan primes are the integers ''R_n'' that are the '''smallest''' to satisfy the condition ...

...lest Gödel numbering schemes is used every day: The correspondence between integers and their representations as strings of symbols. For example, the sequence ...ngs as well. Given a sequence <math>x_1 x_2 x_3 ... x_n</math> of positive integers, the Gödel encoding is the product of the first n primes raised to their co ...

...ven if we worked forever. If a set has the same cardinality as the set of integers, it is called a [[countable set]]. But if a set has the same cardinality a ...d <math>\begin{matrix}\frac{11}{6} \end{matrix} \in \mathbb{Q}</math>. All integers are in this set since every integer ''a'' can be expressed as the fraction ...

...e say whole numbers can be negative. "Positive integers" and "non-negative integers" are another way to include zero or exclude zero, but only if people know t === Integers === ...