[Codeforces Round #641 (Div. 2)]C. Orac and LCM 公式推导

Codeforces Round #641 (Div. 2)C. Orac and LCM

\(\Large{Prove:gcd(lcm(a,b),lcm(a,c))=a*\frac{gcd(b,c)}{gcd(a,b,c)}=lcm(a,gcd(b,c))}\)

\(\Large{gcd(lcm(a,b),lcm(a,c))=gcd(\frac{ab}{gcd(a,b)},\frac{ac}{gcd(a,c)})\\=a*gcd(\frac{b}{gcd(a,b)},\frac{c}{gcd(a,c)})\\=a*gcd(\frac{\prod_{i=1}^{\infty}p_i^{b_i}}{\prod_{i=1}^{\infty}p_i^{min(a_i,b_i)}},\frac{\prod_{i=1}^{\infty}p_i^{c_i}}{\prod_{i=1}^{\infty}p_i^{min(a_i,c_i)}})\\=a*gcd(\prod_{i=1}^{\infty}p_i^{b_i-min(ai,bi)},\prod_{i=1}^{\infty}p_i^{c_i-min(ai,ci)})\\=a*\prod_i^{\infty}p_i^{min(b_i-min(a_i,b_i),c_i-min(a_i,c_i))}\\=a*\prod_i^{\infty}p_i^{min(b_i,c_i)-min(a_i,b_i,c_i)}\\=a*\frac{gcd(b,c)}{gcd(a,b,c)}\\since: \\ if\quad a_i<=b_i\quad and \quad a_i <= c_i\\min(b_i-min(a_i,b_i),c_i-min(a_i,c_i))\\=min(b_i-a_i,c_i-a_i)=min(b_i,c_i)-a_i\\=min(b_i,c_i)-min(a_i,b_i,c_i)\\if \quad a_i>b_i \quad or \quad a_i>c_i\\ min(b_i-min(a_i,b_i),c_i-min(a_i,c_i))\\=min(b_i-b_i,c_i-min(a_i,c_i))\\=0\\=min(b_i,c_i)-min(b_i,c_i)\\=min(b_i,c_i)-min(a_i,b_i,c_i)}\)