if $a\mid x$ and $b\mid x$ and $\gcd(a,b) = 1$, prove $a\cdot b\mid x$
4141
Prove: If $a\mid m$ and $b\mid m$ and $\gcd(a,b)=1$ then $ab\mid m$
157784
Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
84417
Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$
40848
$a\mid b,c\mid d\Rightarrow\,\gcd(a,c)\mid \gcd(b,d)$
27281
How to prove $\,x^a-1 \mid x^b-1 \iff a\mid b$
74561
Let $a_1,a_2,\cdots,a_n$ be $n$ numbers such that $a_i$ is either $1$ or $-1$.If $a_1a_2a_3a_4+\cdots+a_na_1a_2a_3=0$ then prove that $4\mid n$.
21788
Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$
147802
Proving that $\gcd\left(\frac a {\gcd(a,b)},\frac b {\gcd(a,b)}\right) =1$
136304
Prove: $2^b-1\mid{}2^a-1 \iff b\mid{}a$
20823
If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
2289
Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
5789
$ d = \gcd(a,b)\Rightarrow\ \gcd(a/d,b/d) = 1$
23066
If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
7091
If $\gcd(a,b)=1$, then $\gcd(a^n,b^n)=1$
19822
For $D$ a GCD domain , let $a,b,x \in D \setminus \{0\}$ , then is it true that $\gcd (ax,bx)=x \cdot \gcd (a,b)$?
90283
Proof that $3 \mid \left( a^2+b^2 \right)$ iff $3 \mid \gcd \left( a,b\right)$
72424
Suppose 1 ≤ k ≤ n − 1 and gcd(k, n) = 1. Prove that gcd(n − k, n) = 1