Prove the following assertion about modular arithmetic

[Math100]

Homework Statement

Prove the following assertion:
If $a\equiv b \mod n$ and $m\mid n$, then $a\equiv b \mod m$.

Relevant Equations

None.

Proof:

Suppose $a\equiv b \mod n$.
Then $n\mid (a-b)\implies a-b=kn$ for some $k\in\mathbb{Z}$.
Since $m\mid n$, it follows that $n=mp$ for some $p\in\mathbb{Z}$.
Note that $a-b=k(mp)\implies a-b=mq$ where $q=kp$ is an integer.
Thus $m\mid (a-b)$.
Therefore, if $a\equiv b \mod n$ and $m\mid n$, then $a\equiv b \mod m$.

 

[SammyS]

Homework Statement: Prove the following assertion:
If $a\equiv b \mod n$ and $m\mid n$, then $a\equiv b \mod m$.
Relevant Equations: None.

Proof:
Suppose $a\equiv b \mod n$.
Then $n\mid (a-b)\implies a-b=kn$ for some $k\in\mathbb{Z}$.
Since $m\mid n$, it follows that $n=mp$ for some $p\in\mathbb{Z}$.
Note that $a-b=k(mp)\implies a-b=mq$ where $q=kp$ is an integer.
Thus $m\mid (a-b)$.
Therefore, if $a\equiv b \mod n$ and $m\mid n$, then $a\equiv b \mod m$.
$~$

Looks good.