Sort of - if you earnt 13k then you would be taxed down to 10k.
If you earned 13k , then you'd only pay tax on the £250 you've earned above the personal allowance no?
As far as I understand it, it's not that you don't pay taxes so long as you make less than the personal allowance, it's that the first £12750 you make is untaxed regardless of how much you make until you make more than £100k.
What I mean is that you are not taxed on the first £12750 you earn if you are working minimum wage full time, you only pay it on the roughly £6000 you've made after that, which works out to only about £1700 in income tax and national insurance.
So in the example above, they'd have more income than suggested.