# 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).

Prove the transitivity of modular congruence. That is, prove that for all integers a,b,c, and n with n > 1, if .
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The transitivity of modular congruence is that for all integers a, b, c and n with n > 1, if
If , then there exists integers k and k’ such that,
$q-b=nk$
$b-c=n{k}^{\prime }$
$a-c=n\left(k+{k}^{\prime }\right)$
where, k +k’ is an integer.
And so
Conclusion:
The theorem of transitivity of modular congruence is proved.