Chinese Remainder Theorem/Examples/n = 7 mod 12 so n = 3 mod 4

Example of use of Chinese Remainder Theorem

Let $n \equiv 7 \pmod {12}$.

Then:

$n \equiv 3 \pmod 4$

Proof

By the Chinese Remainder Theorem:

$n \equiv 7 \pmod {12} \iff n \equiv 7 \pmod 3 \text { and } n \equiv 7 \pmod 4$

as $3$ and $4$ are coprime.

Thus given the hypothesis:

\(\ds n\)

\(\equiv\)

\(\ds 7\)

\(\ds \pmod {12}\)

\(\ds \)

\(\equiv\)

\(\ds 7\)

\(\ds \pmod 4\)

Chinese Remainder Theorem

\(\ds \)

\(\equiv\)

\(\ds 3\)

\(\ds \pmod 4\)

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Some Properties of $\Z$: Exercise $2.7$