I need to prove: lcm(a,gcd(b,c))=gcd(lcm(a,b), lcm(a,c))
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Is $\operatorname{lcm}(a,\gcd(b,c))=\gcd(\operatorname{lcm}(a,b),\operatorname{lcm}(a,c))$
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$\gcd(a,b)\!=\!1\!=\!\gcd(a,c)\Rightarrow\gcd(a,bc)\!=\!1$ [coprimes to $\,a\,$ are product closed]
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Check: For the integers $a,b,c$ show that $\gcd(a,bc)=\gcd(a,\gcd(a,b)\cdot c)$
85688
How to show $\gcd(ac, bd) = \gcd(a,d) * \gcd(b,c)$
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Proving that $ \gcd(a,b) = as + bt $, i.e., $ \gcd $ is a linear combination.
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GCD to LCM of multiple numbers
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Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
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Prove $\, a_1 \Bbb Z \cap \dotsb \cap a_r \mathbb{Z} = {\rm lcm}(a_1, \ldots, a_r) \Bbb Z$
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Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
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How to prove $\gcd(a^m-b^m,a^n-b^n) = a^{\gcd(m,n)} - b^{\gcd(m,n)} $?
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LCM is associative: $\ \text{lcm}(\text{lcm}(a,b),c)=\text{lcm}(a,\text{lcm}(b,c))$
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Proof that if $\gcd(a,b) = 1$ and $a\mid n$ and $b\mid n$, $ab \mid n$
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Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$
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$a\mid b,c\mid d\Rightarrow\,\gcd(a,c)\mid \gcd(b,d)$
158308
Proving that if $a,b$ are even, then $\gcd(a,b) = 2 \gcd(a/2, b/2)$
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How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?
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Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.