Can I use the 11-digit V5C number to tax a car I've only just bought?

The normal new-keeper route uses the 12-digit reference on the green V5C/2 slip, not the previous keeper's 11-digit log-book number.

Ask the seller for the green slip before leaving and check that the registration matches the vehicle. The main V5C is used by the existing registered keeper for different transactions. If the slip is missing, follow DVLA's instructions for taxing a vehicle without the correct reference rather than attempting to use the seller's details. The seller's road tax ends on transfer, so arrange the buyer's tax before using the vehicle on a public road.