Thread: urgent help needed on proof question!!!!!!

The question has been attached as an image.

any help or light on this question would be hugely appreciated!!

thank you

The first subproblem should be easier to solve in comparison with the second.

let's take a 4 digit number.
Recall that the decimal expansion of any number greater than 10 is of the form ...1000a₄ + 100a₃ + 10a₂ + a₁, where all a_i are positive integers Rearranging the digits of the number will also lead to a similar expansion, say 1000a₃ + 100a₄ + 10a₁ + a₂

Subtraction of the second from the first, or vice versa, will lead to a difference which will be like
900 a₄ + (-900) a₃ + 9a₂ - 9a₁.

It isn't a big surprise that the difference is a multiple of 9.