number theory general formula fisherman problem 4 fisherman catch some amount of fish. If the first fisherman separate the pile into 4 equal amounts excepts with one extra and toss that one extra away. He then takes his 4th and then walks away. The second fisherman then takes the pile that was left behind, divide it into 4 equal except for 1 extra and threw that extra away. He then takes his 4th and then walks away. This happen for the other two fisherman. What is the general formula to determine the total amount of fish in the original pile. An example would be 253. I am having a hard time trying to figure out the formula. 
 A: Each fisherman takes $\frac{1}{4}$ so leaves $\frac{3}{4}$. So if there are $x$ fish after taking their share then beforehand it must be $\frac{4}{3}x+1$. Then repeating this idea 3 more times gives:
$$fish_{initial}=\frac{4}{3}\left(\frac{4}{3}\left(\frac{4}{3}\left(\frac{4}{3}x+1\right)+1\right)+1\right)+1$$
Which gives:
$$fish_{initial}=\frac{256x}{81}+\frac{175}{27}$$
$$=\frac{256x+525}{81}$$
So you need $256x+525$ to divide by 81. In other words:
$$256x+525\equiv0\space(mod\space81)$$
$$13x+39\equiv0\space(mod\space81)$$
From this we can see that $x=-3$ is an obvious solution.
Other solutions will then be in the form of $x=-3+81n, n\in\mathbb{N}$.
Subbing this back in gives:
$$fish_{initial}=256n-3$$
A: [For the following, please refer to the image at the end of this reply. The image is also available in PDF format.]
Here is one way to find the solutions.


*

*Formulate the problem in terms of a liner system of equations.

*Solve the linear system of equations by expressing the original variables in terms of parameter variables.


I am including the following diagram to illustrate the formulation of the linear system of equations based on the problem description, and to present some of the steps in the solution process using an augmented matrix.
I hope this helps. - john

