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I was given the question: A band of 17 pirates has stolen a chest of gold coins. When they try to divide the coins into equal portions, 3 coins are left over.

Could this then be written as $z \equiv 17 \mbox{ mod } 3$?

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    $\begingroup$ It's actually $3\pmod{17}$. One way to think of it is, when you divide the number of coins by $17$, there are $3$ left over. $\endgroup$ – Edward Jiang Nov 12 '15 at 3:50
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You can write it as congruence $$ z \equiv 3 \pmod{17} $$ meaning the left hand side having the same remainder like the right hand side, under division by $17$ or via the modulo operator $$ z \bmod 17 = 3 $$ stating the remainder.

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Did you mean $z \equiv 3 \pmod{17}$?

Yes, of course, meaning that $z$ gives remainder $3$ when divided by $17$.

Therefore you'll have $z=3+17k$ , being $k$ an integer number.

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