You will need both serialnumber and activation code if you install TC on a new machine, else you have problem.
Maybe your old and new pc is a desk model, then you can easily take the hdd from the old one and place in your new pc.
When you did buy TC you get the code's by e-mail or on a disk, so you should try search for that.
V18, V19, V20, V21, 2015/ 16/ 17/ 18 Pro. Platinum
Deluxe V20, V21, 2015/ 16/ 17
Windows 10 Home Premium 64 bit, 32 GB
NVIDIA Geforce GTX 780m, 1 GB