Let m be the least common multiple of a,b and c be another common multiple. Prove that m divides c.
1712
$ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property]
4141
Prove: If $a\mid m$ and $b\mid m$ and $\gcd(a,b)=1$ then $ab\mid m$
76680
Prove that if $\gcd(a,c)=1$ and $\gcd(b,c)=1$ then $\gcd(ab,c)=1$
6410
Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]
257
Prove that : $m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$
53660
Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q  \implies  \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $
13220
find the minimum value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$
21726
Show that if $(b,c)=1$, then $(a,bc)=(a,b)(a,c)$.
26766
Proving : $A \cap (B-C) = (A \cap B) - (A \cap C)$
19896
Prove $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$
150528
Pythagorean triples $(a,b,c)$ and divisibility of $a$ or $b$ by $3$.
2019
Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
136854
Prove: For a,b,c positive integers, ac divides bc if and only if a divides b
55621
Show that if $c_1, c_2, \ldots, c_{\phi(m)}$ is a reduced residue system modulo m, $m \neq 2$ then $c_1 + \cdots+ c_{\phi(m)} \equiv 0 \pmod{m}$
95852
Prove that $(a+b)(b+c)(c+a) \ge8$
147902
Prove $a \mid m$ and $b \mid m \implies \frac{a \cdot b}{\text{gcd}(a, b)} \mid m$
27544
With $a,\ b,\ m,\ n\in \mathbb{Z}$. Show that if ${\rm gcd}(m,n)=1$, then ${\rm gcd}(ma+nb,mn)={\rm gcd}(a,n){\rm gcd}(b,m)$