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Is it true that, given that I have a modulus m, then m is congruent with 0?

My reason for thinking that this is the case is the observation that for any integer k, then

$\ km \ (mod \ m) \cong \ 0 $

Since the residue (remainder) when a number is divided by a multiple of the modulus would be 0.

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    $\begingroup$ Yes, $m\equiv0\pmod{m}$ $\endgroup$
    – saulspatz
    Commented Jun 23, 2018 at 23:36
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    $\begingroup$ "any number k" no, any integer $k$. $\endgroup$
    – Gerry Myerson
    Commented Jun 24, 2018 at 0:40

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$km$ ($mod$ m) means remainder of $km$ divided by $m$.
so $\frac{km}{m}\equiv0$ ($mod$ $m$).

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