how to prove lcm divides any common multiple from definition of lcm(a,b) = (a*b)/gcd(a,b)
4141
Prove: If $a\mid m$ and $b\mid m$ and $\gcd(a,b)=1$ then $ab\mid m$
94770
Greatest common divisor of two positive integers
147802
Proving that $\gcd\left(\frac a {\gcd(a,b)},\frac b {\gcd(a,b)}\right) =1$
128820
Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$
37596
If $\gcd(a,b)= 1$ and $a$ divides $bc$ then $a$ divides $c\ $ [Euclid's Lemma]
130513
LCM is associative: $\ \text{lcm}(\text{lcm}(a,b),c)=\text{lcm}(a,\text{lcm}(b,c))$
117098
Properties of the greatest common divisor: $\gcd(a, b) = \gcd(a, b-a)$ and $\gcd(a, b) = \gcd(a, b \text{ mod } a)$
40538
On determinants and common divisors
135779
Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
11168
Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
101872
Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
11600
$\gcd(a,\operatorname{lcm}(b,c))=\operatorname{lcm}(\gcd(a,b),\gcd(a,c))$
121280
Proving that $a \mathbb{Z} \cap b\mathbb{Z} = \operatorname{lcm}(a,b)\mathbb Z$
21288
For integers $a$ and $b$, $ab=\text{lcm}(a,b)\cdot\text{hcf}(a,b)$
136064
How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?
107914
Expanding GCDs Reasoning: $\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$
45213
Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$