Pirates and Bag of Coins

Question

A group of pirates with 17 members steal a bag of golden coins. When they share their loot evenly, it left with 3 coins. When they argue who should get the remaining of the coins, one of them is killed. Then the coins where shared again evenly. But now it left with 10 remaining coins. Another riot happened and one of them is killed again. This result in dividing the coins again and it turns out all of the pirates have the same amount of coins.

Determine the amount of gold coins in the bag!

Answer 1

Let $k$ be the number of gold coins. Then we have:

$$k \equiv 3 \pmod {17}$$ $$k \equiv 10 \pmod {16}$$

The format is simply $k \equiv \text{number of gold coins remaining} \pmod {\text{number of pirates}}$.

Can you now write the third congruence, and hence use the Chinese Remainder Theorem to find $k$?

Answer 2

$$y\equiv b\bmod m \iff y=mx+b$$ Which means, y has remainder b on division by m (x an integer). We are told the number of coins, y, when divided by 17, gives 3 left ( remainder). This means: $$y=17x+3\iff y\equiv 3\bmod 17$$ Secondly, a person gets killed, leaving 16 people. We are told they now have 10 remaining. The number of coins didn't change so: $$y=16x_1+10\iff y\equiv 16x_1+10$$ The index, is simply to keep potentially different values needed seperate.

Lastly, we are told that another person gets killed. This brings our number of people to 15. It's then said, that this gives no remaining coin:$$y=15x_2+0\iff y\equiv 0\bmod 15$$ Now to combine all this information:

$$15x_2=16x_1+10=17x+3\iff x_2=2x_3\land x=2x_4+1\land x_1=15x_5+5\\30x_3=240x_5+90=34x_4+20\iff x_5=17x_6+16\\4080x_6+3930$$ is our final form. $x_6$ is simply any integer.

The algebra is simpler than putting a geometric twist on it (especially with what I know of MathJax on this site...), as in effect all Chinese Remainder Theorem states is you can join lattice points on the x axis, and equal y values in the integer polynomials to form infinitely many rectangles with predictable side length. This then shows an integer translation can occur to make the two intercept at certain lattice points.