Show that one ampere is equivalent to a flow of 6.25*10^18 elementary charges

It's not. It's equivalent to the transfer of that many elementary charges every second.

The elementary charge (charge on an electron) is

1.60 x 10⁻¹⁹ coulomb / electron.

The number of electrons in a Coulomb is the reciprocal of that number.

6.25 x 10⁻¹⁸ electrons / Coulomb.

There's the number. 1 Coulomb per second is called 1 Ampere.