Editing Schur's theorem (section)

== Combinatorics ==
In [[combinatorics]], '''Schur's theorem''' tells the number of ways for expressing a given number as a (non-negative, integer) linear combination of a fixed set of [[relatively prime]] numbers. In particular, if <math>\{a_1,\ldots,a_n\}</math> is a set of integers such that <math>\gcd(a_1,\ldots,a_n)=1</math>, the number of different multiples of non-negative integer numbers <math>(c_1,\ldots,c_n)</math> such that <math>x=c_1a_1 + \cdots + c_na_n</math> when <math>x</math> goes to infinity is:

:<math>\frac{x^{n-1}}{(n-1)!a_1\cdots a_n}(1+o(1)).</math>

As a result, for every set of relatively prime numbers <math>\{a_1,\ldots,a_n\}</math> there exists a value of <math>x</math> such that every larger number is representable as a linear combination of <math>\{a_1,\ldots,a_n\}</math> in at least one way. This consequence of the theorem can be recast in a familiar context considering the problem of changing an amount using a set of coins. If the denominations of the coins are relatively prime numbers (such as 2 and 5) then any sufficiently large amount can be changed using only these coins. (See [[Coin problem]].)