SOLUTION: Find the number of ordered pairs (a,b) of integers such that \frac{a + 2}{a + 5} = \frac{b}{2}.

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Question 1209133: Find the number of ordered pairs (a,b) of integers such that
\frac{a + 2}{a + 5} = \frac{b}{2}.
Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
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Find the number of ordered pairs (a,b) of integers such that
%28a+%2B+2%29%2F%28a+%2B+5%29 = b%2F2.
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From  %28a+%2B+2%29%2F%28a+%2B+5%29 = b%2F2  we have

      2%2A%28%28a+%2B+2%29%2F%28a+%2B+5%29%29 = b,

      %282a%2B4%29%2F%28a%2B5%29 = b,

       2 - 6%2F%28a%2B5%29 = b,


so  if  "b"  is an integer number, then  6%2F%28a%2B5%29  must be integer.


It means that  a+5  must divide  6  with zero remainder.


Hence, possible values for a+5 are

    a+5 = 6,  a+5 = 3,  a+5 = 2,  a+5 = 1,  a+5 = -1,  a+5 = -2,  a+5 = -3,  a+5 = -6.


Thus we have  8  different integer values for  "a"  that satisfy the condition, 
and, obviously, they produce  8  different ordered pairs of integer numbers (a,b).


So, the ANSWER  to the problem's question is:  there are  8 different pairs (a,b)  of integer number such that  %28a+%2B+2%29%2F%28a+%2B+5%29 = b%2F2.

Solved.