In an ideal transformer:
The magnetic flux remains the same in both the primary and secondary coils because the transformer operates based on mutual induction, where the same magnetic flux links both coils.
The power is conserved in an ideal transformer, meaning the input power (in the primary coil) is equal to the output power (in the secondary coil), ignoring losses.
Thus, the quantities that remain the same are:
\[
\boxed{\text{Magnetic flux and Power both}}.
\]