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Find the number of ordered pairs \((a,b)\) of integers such that

                                                                                                  \( \frac{a + 2}{a + 5} = \frac{b}{4}\)

 Feb 22, 2024
 #1
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Here's how to find the number of ordered pairs (a, b) of integers:

 

Multiply both sides by the common denominator (a + 5) in the first term:

 

a + 2 = b * (a + 5) / 4

 

Simplify and expand the expression on the right:

 

4 * (a + 2) = b * (a + 5)

4a + 8 = ab + 5b

 

Move all terms containing a to one side:

4a - ab = 5b - 8

a(4 - b) = 5b - 8

 

Factor out the greatest common factor (GCF) in both sides:

a(b - 4) = 5(b - 8/5)

 

Analyze the equation:

 

This equation implies that a and (b - 4) are factors of 5.

 

Since a and b are integers, the possible factors of 5 are 1, -1, 5, and -5.

 

Consider each factor combination:

Case 1: a = 1, (b - 4) = 5 - b = 9

Case 2: a = -1, (b - 4) = -5 - b = -1

Case 3: a = 5, (b - 4) = 1 - b = 5

Case 4: a = -5, (b - 4) = -1 - b = -3

 

Check for solutions:

 

All four cases lead to valid integer solutions for a and b: (1, 9), (-1, -1), (5, 5), and (-5, -3).

 

Therefore, there are 4 ordered pairs (a, b) of integers that satisfy the given equation.

 Feb 22, 2024

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