Let $S$ be the set of all integers of the form $ax + by$. Suppose that $S$ is not empty and let $d = ax_0 + by_0$ be the least element in $S$.
56612
Proof of $\gcd(a,b)=ax+by\ $ [Bezout's identity]
134159
Smallest value of $a^2 + b^2 + c^2+ d^2$, given values for $(a+b)(c+d)$, $(a+c)(b+d)$, and $(a+d)(b+c)$
152693
If $C|a$ and $C|b$ then $C|(ax+by)$
100565
Property of sup of a set of numbers
136755
Direct proof of empty set being subset of every set
3694
Show $GCD(a_1, a_2, a_3, \ldots , a_n)$ is the least positive integer that can be expressed in the form $a_1x_1+a_2x_2+ \ldots +a_nx_n$
22154
Show that given a set of positive n integers, there exists a non-empty subset whose sum is divisible by n
103744
Minimal $ab$ for Rational Number $a/b$ in an Interval
7785
Cardinality of sum-set
126015
Elementary number theory in sets
106729
Least Value Of $x+y+z$
122271
A subset of a transitive set is transitive
103717
Topological Property of the Set of Integers
124325
Element of, subset of and empty sets
79568
Cardinality of the set $D$
103314
Let $\{K_n \}_{n \in \mathbb{N}}$ be a collection of sets, where $K_n = [n, \infty)$. Why is $\bigcap_{n \in \mathbb{N}} K_{n}$ empty?
44076
Proving if $d_0$ is the smallest positive integer in $S$ then $d_0 = \gcd(a,b)$