Prove: if $a \mid c$ and $b \mid d$ then $ab \mid cd$
111800
If $a|c$ and $b|d$, is it true that $ab|cd$?
22845
Prove that if d is a common divisor of a and b, then $d=\gcd(a,b)$ if and only if $\gcd(a/d,b/d)=1$
24845
Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
157784
Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
52565
Prove that if $d \mid n \in \mathbb{N}$, then $\varphi(d) \mid \varphi(n)$.
111003
Prove that if $A\triangle B = C\triangle B$, then $A = C$
30583
If $a\mid b$ and $b\mid a$, $a,b\in R$, show that $\exists w \in R^\times$ with $b=wa$
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 $ab|cd$.
21788
Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$
147902
Prove $a \mid m$ and $b \mid m \implies \frac{a \cdot b}{\text{gcd}(a, b)} \mid m$
50266
In the following figure prove that: $AK||BC$
77164
If $a \mid m$ and $(a + 1) \mid m$, prove $a(a + 1) | m$.
50614
If $4\mid a+bc$ and $6\mid b+ac$ prove that $2\mid a^2-b^2$
135779
Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
80789
Prove that $\forall a,b,c,d\in \mathbb{R}\,$, $(a^2-b^2)(c^2-d^2)\leq(ac-bd)^2$ and equalities hold if and only if $ad=bc$.
39217
If $d \mid n$ then $x^{q^d} - x \mid x^{q^n} - x$.
152693
If $C|a$ and $C|b$ then $C|(ax+by)$