A fundamental example in number theory is descent by (Euclidean) division with remainder (or, equivalently, by repeated subtraction), as in the following basic result.

**Lemma** $\ \ $ Let $\,S\,$ be a nonempty set of positive integers that is closed under subtraction $> 0,\,$ i.e. for all $ \,n,m\in S, \,$ $ \ n > m\ \Rightarrow\ n-m\, \in\, S.\,$ Then the least $ \:\ell\in S\,$ divides every element of $\, S.$

**Proof** ${\bf\ 1}\,\ $ If not there is a least nonmultiple $ \,n\in S,\,$ contra $ \,n-\ell \in S\,$ is a nonmultiple of $ \,\ell.$

**Proof** ${\bf\ 2}\, \,\ \ S\,$ closed under subtraction $ \,\Rightarrow\,S\,$ closed under remainder (mod), when it is $\ne 0,$ since mod is simply repeated subtraction, i.e. $ \ a\bmod\ b\, =\, a - k b\, =\, a\!-\!b\!-\!b\!-\cdots\! -\!b.\,$ Therefore $ \,n\in S\,$ $\Rightarrow$ $ \, (n\bmod \ell) = 0,\,$ else it is in $\, S\,$ and smaller than $ \,\ell,\,$ contra minimality of $ \,\ell.$

**Remark** $\ $ In a nutshell, two applications of induction yield the following inferences

$\begin{eqnarray}\rm S\ closed\ under\ {\bf subtraction} &\:\Rightarrow\:&\rm S\ closed\ under\ {\bf mod} = remainder = repeated\ subtraction \\
&\:\Rightarrow\:&\rm S\ closed\ under\ {\bf gcd} = repeated\ mod\ (Euclid's\ algorithm) \end{eqnarray}$

This yields Bezout's GCD identity: the set $ \,S\,$ of integers of form $ \,a_1\,x_1 + \cdots + a_n x_n,\ x_i\in \mathbb Z,\,$ is closed under subtraction so Lemma $\Rightarrow$ every positive $ \,k\in S\,$ is divisible by $ \,d = $ least positive $ \in S.\,$ Therefore $ \,a_i\in S$ $\,\Rightarrow\,$ $ d\mid a_i,\,$ i.e. $ \,d\,$ is a *common* divisor of all $ \,a_i,\,$ necessarily the *greatest* such because $ \ c\mid a_i$ $\Rightarrow$ $ \,c\mid d = a_!\,x_1\!+\!\cdots\!+\!a_nx_n$ $\Rightarrow$ $ \,c\le d.\,$ When interpreted constructively, this yields the extended Euclidean algorithm for the gcd.