further to what Klarzy said. If the portions of V5 have been returned to the DVLA dated with 24th August, and you wish to use it straight away, you will have to tax it from the 1st of August. The previous owner, if they have paid in advance for tax, they will receive a refund for the remaining full months. ie from September onwards. Govt. get a double month of tax.
However, if you can refrain from using it, (or you wish to take a chance on not being spotted - not condoning this!)you can SORN it straight away, and then tax it on the 1st of September. Doing it this way, the govt only get 7 days of tax (24th - 31st Aug) unnecessarily, and you dont get stung for 24 days worth.
It has been discussed not only on this forum, but on many other forums and elsewhere. If they use a computer network to obtain the tax from you, when you either retax the car, or purchase the car, then surely they can take a payment for a part month!
to claim it is for administration reasons, I find laughable (I was told this by phone) since,
1. it is unfair to some car owners (different values of tax)
2. there is minimal administration if any at all, when the keeper is the one doing the administrating!!!
tax = VED