$Ax+By+Cz=D \text { has a solution iff } \gcd(\gcd(A,B),C)\mid D$
57129
How to find an integer solution for general Diophantine equation $ax + by + cz + dt... = N$
63758
$c\mid ab\Rightarrow c\mid \gcd(a,c)\gcd(b,c),\,$ for integers $a, b, c$
24845
Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
946
How to find solutions of linear Diophantine ax + by = c?
32463
$ax\equiv b\pmod{\! n}$ has a unique solution if $\gcd(a,n) = 1$
25825
Proof that $\gcd(ax+by,cx+dy)=\gcd(x,y)$ if $ad-bc= \pm 1$
158308
Proving that if $a,b$ are even, then $\gcd(a,b) = 2 \gcd(a/2, b/2)$
157784
Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
124320
$a,b,c>0 \text{ s.t. }a+b+c=1 \implies \sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ca+b} \ge 1+ \sqrt{ab}+\sqrt{bc}+\sqrt{ca}$
128820
Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$
97667
If $a<b$, $c<d$ then $a+c<b+d$?
118809
If $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c
152342
$x_1+x_2+\cdots+x_n\leq M$: Cardinality of Solution Set is $C(M+n, n)$
4486
If $ar + bs =1$, then $\gcd(a,s) = \gcd(r,b) = \gcd(r,s) = 1$
120292
How to formally prove: if $d\mid da+b$, then $d\mid b$?
147527
Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = \gcd(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$.