I have for the first time in my life tried to write down a conjecture which I undoubtedly assume has a nice and understandable proof, but I kind of wanted to see if it’s true by myself or if I’m wrong. Please help me out.
“Given a set $X$ with elements $x$ in it, represented in terms of remainders$\mod n$ as such: $x = q\cdot n + b$; the following congruence holds: The product of all the elements$\mod n$ is congruent to the product of all the remainders $b\mod n$”
This might seem both intuitive and childishly easy to many of you, but I’d love some guidance with a rigorous proof.
Thank you!
Prove it for two numbers, say:
$$\begin{cases}x=mn+a=a\pmod n\\{}\\y=ln+b=b\pmod n\end{cases}\;\;\implies xy=mln^2+(al+bm)n+ab=ab\pmod n$$
Now use a little induction...
This stuff is studied in basic algebra, either within number theory, or m,ore likely perhaps in group theory...