Question 1209235
<pre>

{{{mn = 3m + 3n + 17}}}
{{{mn - 3m = 3n + 17}}}
{{{m(n-3)=3n+17}}}
{{{m=(3n+17)/(n-3)}}}

Long division:

   <u>     3</u>
n-3)3n+17
    <u>3n- 9</u>
       26

{{{m}}}{{{""=""}}}{{{3+26/(n-3)}}}

n-3 must divide evenly into 26, so n-3 is either ±1,±2,±13, or ±26

That's 8 possibilities, so we know the answer is 8. To check, we find them:

1. if n-3 = -26, then n = -23, and m = 3+26/(-23-3) = 2, ordered pair (2,-23).
2. if n-3 = -13, then n = -10, and m = 3+26/(-10-3) = 1, ordered pair (1,-10).
3. if n-3 = -2, then n = 1, and m = 3+26/(1-3) = -10, ordered pair (-10,1).
4. if n-3 = -1, then n = 2, and m = 3+26/(2-3) = -23, ordered pair (-23,2).
5. if n-3 = 1, then n = 4, and m = 3+26/(4-3) = 29, ordered pair (29,4).
6. if n-3 = 2, then n = 5, and m = 3+26/(5-3) = 16, ordered pair (16,5).
7. if n-3 = 13, then n = 16, and m = 3+26/(16-3) = 5, ordered pair (5,16).
8. if n-3 = 26, then n = 29, and m = 3+26/(29-3) = 4, ordered pair (4,29).


So indeed, there are 8 such ordered pairs.

Here is a plot of all 8.  See how nicely they fit on that curve.

Can you guess what the equation of that curve is?

{{{drawing(300,300,-31,31,-31,31, graph(300,300,-32,32,-32,32,(3x+17)/(x-3)),

circle(2,-23,0.5),
circle(1,-10,0.5),
circle(-10,1,0.5),
circle(-23,2,0.5),
circle(29,4,0.5),
circle(16,5,0.5),
circle(5,16,0.5),
circle(4,29,0.5)

)}}}
Edwin</pre>