An intuitive explanation of the multiplication identity of modular arithmetic Is there any intuitive explanation for the validity of this identity?

If $a\equiv b\pmod n$ and $c\equiv d\pmod n$, then
$$a\times c \equiv b\times d \pmod n$$

I want something which appeals to someone's who just beginning to learn number theory.

This is not difficult to prove and understand. The idea is to use a simpler result two times.

If $a\equiv b\pmod{n} $ then we have $ka\equiv kb\pmod{n} $.

This should be obvious to understand as it is an immediate consequence of the definition of congruence. Just note that if $a-b$ is a multiple of $n$ then $k(a-b) $ is also a multiple of $n$.
Using this we can multiply first congruence with $c$ and second congruence with $b$ to get $$ac\equiv bc\pmod{n}, bc\equiv bd\pmod {n} $$ and these two give you $ac\equiv bd\pmod{n} $ (this is based on the fact that if two numbers are multiple of $n$ then so is their sum).

I don't think the above would count as a lot of hairy algebra and ideally should be accessible to anyone who has basic idea of factors and multiples (typically 6th standard math suited for children of 10-11 years in Indian curriculum). 
