If $(a,b)=1$ and $a|c$ and $b|c$ then $ab|c$
93686
In a UFD, lcm = product for coprimes: $\,(a,b)=1,\ a,b\mid c\Rightarrow ab\mid c$
20823
If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
41099
If $A \subseteq C$ and $B \subseteq D$ then $A \times B \subseteq C \times D$
24845
Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
111800
If $a|c$ and $b|d$, is it true that $ab|cd$?
62474
If $AB=I_n$ and $BA=I_m$ then $n=m$.
118239
If $A, B, C$ are sets prove that $(A\setminus B)\setminus C\subseteq A\setminus(B\setminus C)$
124868
Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
152693
If $C|a$ and $C|b$ then $C|(ax+by)$
142993
Proof of A ⊆ C and B ⊆ C ⇔ A ∪ B ⊆ C
147246
If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$
35242
If $a$, $b$, $c$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$, and $\frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ are all integers, then $|a|=|b|=|c|$
113211
Arithmetics of cardinalities: if $|A|=|C|$ and $|B|=|D|$ then $|A\times B|=|D\times C|$
6096
If $A+B=AB$ then $AB=BA$
26861
$a+b=c \times d$ and $a\times b = c + d$
72092
If $A^2 = I$, $B^2=I$ and $(AB)^2=I$, then $AB = BA$
90438
To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$