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I am trying to prove relation between G.C.D and L.C.M for three integers. Is the following relation even true. If yes, then I need hints please. let a, b and c be any integers then,

$$lcm(a,gcd(b,c))=gcd(lcm(a,b),lcm(a,c))$$ Or in notations, $$[a,(b,c)]=([a,b],[a,c])$$

Abdul Mujeeb
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  • Have you tried some examples? – Robert Israel Dec 31 '20 at 16:32
  • @RobertIsrael Yes, this holds in examples. But I need to prove it generally. – Abdul Mujeeb Dec 31 '20 at 16:36
  • We can write $a = \prod_{i=1}^\infty p_i^{k_i}$ (where an infinite number of $k_i$ are zero and only a finite number of them aren't zero) and $b = \prod_{i=1}^\infty p_i^{m_i}$ then $\gcd(a,b) = \prod_{i=1}^{\infty} p_i^{\min(k_i,m_i)}$ and $lcm(a,b) = \prod_{i=1}^{\infty} p_i^{max(k_i, m_i)}$. If you use that this becomes very easy. – fleablood Dec 31 '20 at 17:15

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Hint:

Try using the fundamental theorem of arithmetic, but write with all prime numbers (some with power $0$) rather than just the ones in the factorization.

Then if you look at each prime number, you will find the problem becomes essentially $max(A, min(B,C))=min(max(A,B), max(A,C))$ which I will leave you to prove.

cr001
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