why $ax + by$ can take any value if their gcd is $1$
63651
If $(a,b)=1$ then there exist positive integers $x$ and $y$ s.t $ax+by=1$.
422
If $\gcd(a,b)=1$, $\gcd(a,y)=1$ and $\gcd(b,x)=1$ then prove that $ax+by$ is prime to $ab$
152494
How is  $Ax + By = C$ the equation of a straight line?
22845
Prove that if d is a common divisor of a and b, then $d=\gcd(a,b)$ if and only if $\gcd(a/d,b/d)=1$
62342
$x_1 \cdot \dots \cdot x_n=1 \implies x_1 + \dots + x_n \ge n$
145354
What is $\gcd(0,0)$?
136064
How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?
20823
If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
113641
Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c)
105832
prove that if $\gcd(a,b)=1$ then $\gcd(a^2, b^2)=1$
56423
gcd of power plus one
147079
$\gcd\left(\binom{n}{k},n\right)>1$ for nontrivial values?
23066
If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
151434
If integers $a$ and $b$ are not both zero, then $\gcd(a,b)=\gcd(a-b,b)$
152693
If $C|a$ and $C|b$ then $C|(ax+by)$
1713
number theory $\gcd(a,bc)=\gcd(a,c)$
37596
If $\gcd(a,b)= 1$ and $a$ divides $bc$ then $a$ divides $c\ $ [Euclid's Lemma]