# How many times does this expression give the value 0 as modulus?

$S=\{1,2,3\ldots,19\}$

$(5k + 5) \mod 20$

$\gcd(20,5) = 5$

$20$ and $5$ are divisible by $5$ and $1$.

thus the expression gives the value $0$ $2$ times between $1$ and $19$?

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As you have been posting a few questions to this site, I would encourage you to learn how to format them so the math looks nice. See meta.math.stackexchange.com/questions/5020/… and/or meta.math.stackexchange.com/questions/1773/… – Gerry Myerson Oct 28 '12 at 22:47

This problem is small enough that you can just try all values of $k$, and when you do, you find $k=3$ works (that is, $5k+5\equiv0\pmod{20}$ if $k=3$), and also $k=7$, $k=11$, $k=15$, and $k=19$.