The two numbers whose arithmetic mean is 34 and their geometric mean is 16 are:

64, 4

Take the help from options =

If we choose option = 3

So, the numbers are = 64, 4

arithmetic mean = $\frac{64 + 4}{2}$ = 34

and geometric mean = $\sqrt {64 × 4}$ = 16

Both the values of arithmetic mean and geometric mean are satisfying the given values. So the numbers are = 64, 4