MHB Is My Solution to Part B of the Math Question Correct?

[Joe20]

Hi, I have attached the question and the solutions to part a and b of this question. Would like someone to verify if I have done anything wrong especially with part b. Greatly appreciate it! Thanks.

 

[Euge]

Your solution is correct. It could be simplified greatly by noting $S = 2\Bbb Z_{10}$. So, for instance, in proving closure under addition, you could the following: if $x, y\in S$, with $x = 2m$ and $y = 2n$, then $x + y = 2(m + n)\in S$.