+24 Find the number of ordered pairs \((a,b)\) of integers such that
\( \frac{a + 2}{a + 5} = \frac{b}{4}\)
+1610 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.
