number theory lcm

Question

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.

Answer 1

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 $a,b,c$