Question $1$: $[15$ points$]$

$($a$)[2$ points$]$ Discuss why ubrace$(100.\mathrm{.}0$ubrace${)}_m-=1(mod9)$ for every $m2$

$($b$)[3$ points$]$ Prove that the remainder of any number modulo $9$ is equal to the remainder of the sum of its digits modulo $9.$ As an example

$15231-=1+5+2+3+1-=12-=3,(mod9)$

$($You can also try to come up with a method to find the remainder of a number modulo $11$ using similar type of ideas. No need to write it down, but try to solve it $)$

$($c$)[10$ points$]$ Imagine we write the numbers $1,2,$cdots,$2024$ from left to right to create the following number

$x=12345678910111213$cdots$20232024$

What is the remainder of this number modulo $9.$

Hint: You might try to add the digits but this will be really hard and is not a good idea here. Because then you have to calculate how many times each digit appears which is not easy. You might want to use similar type of ideas, but smarter $=)$