Question 1209235
<br>
{{{mn=3m+3n+17}}}<br>
Solve the equation for one variable in terms of the other:<br>
{{{mn-3m=3n+17}}}
{{{m(n-3)=3n+17}}}
{{{m=(3n+17)/(n-3)}}}<br>
Perform the indicated division and express the result as quotient and remainder:<br>
{{{m=((3n-9)+26)/(n-3)}}}
{{{m=3+26/(n-3)}}}<br>
In that last equation, m and 3 are integers, so {{{26/(n-3)}}} must be an integer.<br>
The number of ordered pair solutions is the number of integer factors of 26, which is 4.<br>
NOTE: In typical problems like this, we are looking for solutions in positive integers.  However, this problem does not specify positive integers; counting positive and negative integers, the number of integer factors of 26 is 8.<br>
So there are 8 ordered pair solutions.<br>
ANSWER: 8<br>
The problem doesn't ask us to find the solutions, but we can do so to verify that there are 4 pairs of solutions.  Note that the expression is symmetrical in m and n, so if (a,b) is a solution the (b,a) will be a solution.  So to find the 8 solutions we only need to find 4 of them and switch the order of the two numbers to get the other solutions.<br><pre>

   n-3  n  m=3+26/(n-3)   solutions (m,n)
  ---------------------------------------
    1   4   3+26/1 = 29     (4,29) and (29,4)
    2   5   3+26/2 = 16     (5,16) and (16,5)
   -1   2   3+26/-1 = -23   (2,-23) and (-23,2)
   -2   1   3+26/-2 = -10   (1,-10) and (-10,1)</pre>