Python 3.9, 44 bytes

lambda l:(gcd(*l),lcm(*l))
from math import*

In Python 3.8 and lower, gcd and lcm only accept 2 arguments. This was changed in 3.9, so this answer only works on 3.9+. Try it online does not have the needed version, so I could not post a link to it.

Ruby, 35 bytes

->l{[:gcd,:lcm].map{|x|l.reduce x}}

Try it online!

Numlang, 8 bytes

Code

$;Gy;OCy

Explanation

$;Gy;OCy
$;          // Take input from user (implicitly store in y)
  Gy;       // Get the GCD of y (implicitly store in l)
     O      // Output the last used variable (l)
      Cy    // Get the LCM of y (implicitly store in l)
            // Implicitly output the last used variable (l)