Five-digit number

Anna thinks of a five-digit number that is not divisible by three or four. If he increments each digit by one, it gets a five-digit number that is divisible by three. If he reduces each digit by one, he gets a five-digit number divisible by four. If it swaps any two digits, the number is lower. John claims there are more than five possibilities. Was he right?

Correct result:

n =  6

Solution:

$$\begin{gathered} n_1=65431 \\ {a}_1=76542 \\ {b}_1=54320 \\ {n}_2=76531 \\ {a}_2=87642 \\ {b}_2=65420 \\ {n}_3=76543 \\ {a}_3=87654 \\ {b}_3=65432 \\ {n}_4=86431 \\ {a}_4=97542 \\ {b}_4=75320 \\ {n}_5=87631 \\ {a}_5=98742 \\ {b}_5=76520 \\ {n}_6=87643 \\ {a}_6=98754 \\ {b}_6=76532 \\ n=6 \end{gathered}$$

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