I'm having a hard time understanding modular numbers and aritmetic.

If I have $\displaystyle r=a(\mod P)$, then $\displaystyle r-a=bP$ where $\displaystyle b$ is an integer. So if I want to find 23 in $\displaystyle \mod 12$, I write $\displaystyle r=23(\mod 12)$ which should turn out to be 11 right? So 23 is equal to 11 in $\displaystyle \mod 12$.

So if I want to find:

$\displaystyle r=26(\mod 12)$

I need to find r such that:

$\displaystyle r-26=b12$

where b is an integer and r is the smallest possible positve number satisfying the equation. Is this correct? 26 hours would bring us to 2 o-clock since 2 is the smallest positive integer such that $\displaystyle 2-26=b12$ where $\displaystyle b$is an integer. So here, $\displaystyle b=-2$.

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One more example with a different mod that isn't so familiar:

$\displaystyle r=3(\mod 7)$

$\displaystyle r-3=b7$

So the smallest possible number seems to be 10 with $\displaystyle b=1$.

$\displaystyle 10=3(\mod 7)$

where b is an integer.

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Is my reasoning correct?