A bus goes to 3 bus stops, at each stop 3/4 of the people on the bus get off and 10 get on. what is the minimum number of people to start on the bus? 
A bus goes to $3$ bus stops, at each stop $3/4$ of the people on the bus get off and $10$ get on. what is the minimum number of people to start on the bus?

I think the number would need to be divisible by $4$ and an integer, since you can't have a non "full person".
I assumed that the people on the bus do NOT include the driver.
What I have so far:
let $"n" = \#$ of people on the bus.
First stop: $n/4 + 10$
Second stop: $(n/16+10/4) +10$
Third stop: $(n/64 + 50/16) +10= (840+n)/64$
Don't know how to move on from here to solve... and how do I account for the amount of people who left the bus?
Please help! Thanks!
 A: Let $R \equiv \{0, 1, 2, \cdots \}, \;S \equiv \{1, 2, 3, \cdots \}.$
Initially, there are $x_0$ people.
After the 1st stop, there are $x_1$ people.
After the 2nd stop, there are $x_2$ people.
After the 3rd stop, there are $x_3$ people.
(1) $\;x_0$ goes to $(1/4)x_0 + 10 = x_1.$
(2) $\;x_1$ goes to $(1/4)x_1 + 10 = x_2.$
(3) $\;x_2$ goes to $(1/4)x_2 + 10 = x_3.$
It is immediate that $x_0, x_1, x_2$ are all multiples of 4
$\;\Rightarrow$ 
$\exists \;a,b,c \,\in \,S \;\ni $
$\; x_0 = 4a, \; x_1 = 4b, \; x_2 = 4c.$
(4) By (2), $\;b + 10 = 4c \;\Rightarrow\; c \geq 3 \;\Rightarrow\;
\exists \;k \,\in \,R \;\ni c = 3 + k \;\Rightarrow $ 
$x_2 = (12 + 4k) \;\Rightarrow$
[by (2)] $\;b = (x_2 - 10) = 2 + 4k \;\Rightarrow $ 
$x_1 = (8 + 16k) \;\Rightarrow$ 
[by (1)] $\;(1/4) x_0 = a = (x_1 - 10) = [(8 + 16k) - 10] = 16k - 2 \;\Rightarrow$
$k \geq 1\;$ and $\;x_0 = 4a = 64k - 8 \;\Rightarrow$
The min value for $x_0$ is 56.
Addendum
Originally, I thought that the answer was 40.
Then I realized that I was misreading the query.  That is, at each stop, 3/4 of the people get off (before 10 get on), not 1/4 of the people.
Addendum-1
A fair criticism of my answer is that I didn't try to focus on the OP's work, and guide his work to a solution.  I neglected to try, because with a problem like this I'm only comfortable taking baby steps, so I'm uncomfortable trying to critique a sophisticated approach.
A: You found out that when $n_0\geq1$ passengers are on the bus at the start then after three stops there are
$$n_3={840+n_0\over64}$$
people on the bus. As $n_3$ has to be an integer the smallest $n_0$ that would qualify is $n_0=56$, making $n_3=14$. In order to make sure we have to check that for this $n_0$ the intermediate numbers $n_1$ and $n_2$ are integers as well.
By the way: When $x_k$ is the number of passengers after $k$ stops then we have the recursion
$$x_{k+1}={1\over4} x_k+10\ .$$
The "Master Theorem" gives the general solution
$$x_k=c\cdot 4^{-k}+{40\over3}\qquad(k\geq0)\ ,$$
but this expression does not care about integerness. Therefore we really have to go through the cases.
A: Now expand your third expression to put everything over a common denominator.  For example at the third stop you have $\frac{stuff}4+7=\frac {stuff+28}7$ $stuff$ still has fractions in it, so unpack them.  See what the denominator comes out to be and see what the smallest $n$ is to make the fraction an integer.
A: Let the bus have x passengers in the beginning
At first stop:
No. of people left in the bus=  x/4 + 10
At second stop:
No. of people left on the bus=  x/16 + 50/4
At third stop:
No. of people left on the bus=  x/64 + 210/16
Hence, the number of people left on the bus after the bus stops three times is (840 + x)/64
Obviously, the number of people can neither be fractional, nor negative
Hence, 840+x has to be a multiple of 64.
The minimum possible value of x for which 840+x is a multiple of 64 is   64*14-840
Hence x=56
