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If $a,b,c$ are any three positive integers, Is it true that $HCF(HCF(a,b),c)=HCF(a,HCF(b,c))$

My try:

Case $1.$ if atleast one of $a,b,c$ is equal to ONE , then the claim is True.

case $2.$ If every pair is Relatively Prime then the Claim is True

But i will get lot of cases? Any HINT to prove or any counter example to disprove the claim?

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Let $a=\prod_j p_j^{i_{j,a}}$, $b=\prod_j p_j^{i_{j,b}}$, $c=\prod_j p_j^{i_{j,c}}$ be the prime factorization.

We have $HCF(a,b)=\prod_j p_j^{\min(i_{j,a}, i_{j,b})}$.

and $HCF(HCF(a,b),c)=\prod_j p_j^{\min(\min(i_{j,a}, i_{j,b}), i_{j,c})}=\prod_j p_j^{\min((i_{j,a}, i_{j,b}, i_{j,c})}$

Notice that we have $\min(\min(m,k),l)=\min(m,\min(k,l)).$

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