<PRE><B>Question 1194075</B>
&lt;br&gt;
x = number of stops where the bus picks up 5 people
y = number of stops where the bus picks up 2 people and lets off 5&lt;br&gt;
At each of the x stops, the number of people on the bus increases by 5.
At each of the y stops, the number of people on the bus decreases by 3.&lt;br&gt;
Then after (x+y) stops the number of people on the bus is 5x-3y.&lt;br&gt;
We are to find the least number of total stops (x+y) greater than 17 for which the number of people left on the bus is 11:&lt;br&gt;
{{{5x-3y=11}}} ; subject to the constraint that x+y&gt;17&lt;br&gt;
This is a linear Diophantine equation -- a single equation with two variables, in which the values of the variables are integers.&lt;br&gt;
Here is a standard formal way for solving this kind of equation.&lt;br&gt;
(1) Solve the equation for one of the variables (it doesn&#39;t matter which)&lt;br&gt;
{{{5x-3y=11}}}
{{{5x=3y+11}}}&lt;br&gt;
Divide by 5, writing the result on the right as quotient plus remainder:&lt;br&gt;
{{{x = (3y+11)/5 = (10+(3y+1))/5 = 2+(3y+1)/5}}}&lt;br&gt;
(2) Find the values of y that make that expression an integer; find the corresponding values of x, and find the value of x+y.  We are looking for the smallest value of x+y that is greater than 17.&lt;br&gt;&lt;pre&gt;
    y   x=2+(3y+1)/5  x+y
  ------------------------
    3   2+2 = 4        7
    8   2+5 = 7       15
   13   2+8 = 10      23&lt;/pre&gt;
ANSWER: 23&lt;br&gt;
A note about solving linear Diophantine equations by this method....&lt;br&gt;
Note that in the table above, the values of x and y (and x+y) form arithmetic sequences. Specifically, the possible y values have a common difference of 5 and the possible x values have a common difference of 3.  The &quot;5&quot; and &quot;3&quot; are because of the equation x=(3y+1)/5.&lt;br&gt;
If you are solving a problem like this and your list of x and/or y values does not form an arithmetic sequence, then some of your calculations are incorrect.&lt;br&gt;
At the same time, knowing that these lists of values form arithmetic sequences allows you to find other solutions without searching for x or y values.  For example, in this problem, seeing that the first two possible y values are 3 and 8, you know that the common difference is 5, so the next possible y values are 13, 18, 23, ...; and since the first two possible x values are 4 and 7, you now that the next possible corresponding x values are 10, 13, 16....&lt;br&gt;
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