Is it true that $HCF(HCF(a,b),c)=HCF(a,HCF(b,c))$

Question

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?

Does this answer your question? How to prove $\gcd(a,\gcd(b, c)) = \gcd(\gcd(a, b), c)$? — John Omielan, Jun 30 '20 at 04:28
Note there are several other closely related questions on this site, such as the somewhat more general one of Number Theory Prove $\gcd(a, b, c)=\gcd(\gcd(a,b),c)=\gcd(\gcd(a,c),b)=\gcd(\gcd(b,c),a)$.. — John Omielan, Jun 30 '20 at 04:31

Siong Thye Goh · Answer 1 · 2020-06-30T03:55:23.537

1

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 $\min(\min(m,k),l)=\min(m,\min(k,l)).$

edited Jun 30 '20 at 03:55

answered Jun 30 '20 at 03:44

Siong Thye Goh

1 Answers1