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prove if $a, b ,c$ are positive integers, then $lcm(a,lcm(b,c))=lcm(a,b,c)=lcm(lcm(a,b),c)$
$lcm$ is least common multiple
My thought is to show that they have common divisors but not sure how to go about it.

cap
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  • Observe that any common multiple of $a$ and $b$ is a multiple of $\text{lcm}(a,b)$. – Angina Seng Nov 27 '18 at 06:54
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  • https://math.stackexchange.com/questions/254704/prove-that-textlcm-textlcma-b-c-textlcma-textlcmb-c – 1ENİGMA1 Nov 27 '18 at 07:33

1 Answers1

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Hint:

If the highest exponents of prime $p$ in $a,b,c$ be $A,B,C$ respectively.

WLOG $A\ge B\ge C$

Now the highest exponents of prime $p$ in lcm$(a,b,c)$ will be max$(A,B,C)=A$

lcm$(a,$lcm$(b,c))=$max$(A,$max$(B,C))=$max$(A,B)=A$

lcm$($lcm$(a,b),c))=$max$(A,C)=A$

This holds true any prime that divides at least one of $a,b,c$

lab bhattacharjee
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