Prove the transitivity of modular congruence. That is, prove that for all integers a,b,c, and n with n > 1, if a = b(mod n) and b = c(mod n) then a = c(mod n).

The transitivity of modular congruence is that for all integers a, b, c and n with n > 1, if $a\equiv b(modn)andb\equiv c(modn)thena\equiv c(modn)$
If $a\equiv b(modn)andb\equiv c(modn)$, then there exists integers k and k’ such that,
$q-b=nk$
$b-c=n{k}^{\prime }$
Adding these two equations yields
$a-c=n(k+k^{\prime })$
where, k +k’ is an integer.
And so $a\equiv c(modn)$
Conclusion:
The theorem of transitivity of modular congruence is proved.