Prove that $\operatorname{lcm}(a,b,c)=\frac{abc \operatorname{gcd}(a,b,c)} {\operatorname{gcd}(a,b)\operatorname{gcd}(b,c)\operatorname{gcd}(c,a)}$
121156
Prove that: $\gcd[a,b,c]=\frac{abc.\operatorname{lcm}(a,b,c)}{\operatorname{lcm}(a,b)\operatorname{lcm}(a,c)\operatorname{lcm}(b,c)}$
37346
If $m, n$ be positive integers, prove that $\phi(mn)=\phi((m,n))\phi([m,n])$, where $(m,n)=$ gcd of $m, n$ and $[m, n]=$ lcm of $m, n$.
23694
Given that $abc=1$, prove that $\frac{b}{c} + \frac{c}{a}+ \frac{a}{b} \ge \frac{1}{a} + \frac{1}{b} +\frac{1}{c}$.
118809
If $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c
147246
If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$
70951
Prove divisibility law $\,b\mid a,c\,\Rightarrow\, b\mid ka + lc$ for all $k,l\in\Bbb Z$
50396
How can I show that $ab \sim \gcd (a,b) {\operatorname{lcm} (a,b)}$ for any $a,b \in R \setminus \{0\}$?
45213
Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$
10348
Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
93686
In a UFD, lcm = product for coprimes: $\,(a,b)=1,\ a,b\mid c\Rightarrow ab\mid c$
99376
Prove that If $m'$ is a common multiple of $s$ and $t$, then $m | m'$. Here $m$ is the LCM of $s$ and $t$.
76680
Prove that if $\gcd(a,c)=1$ and $\gcd(b,c)=1$ then $\gcd(ab,c)=1$
60777
Prove $\, a_1 \Bbb Z \cap \dotsb \cap a_r \mathbb{Z} = {\rm lcm}(a_1, \ldots, a_r) \Bbb Z$
111555
Find $a,b \in \mathbb{N}$ with $\mathrm{lcm}(a,b)=12\gcd(a,b)\,$ and $\,a\bmod b = 5$
23066
If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
40848
$a\mid b,c\mid d\Rightarrow\,\gcd(a,c)\mid \gcd(b,d)$
50249
Pairs of numbers with a given LCM